Local inequalities for $cA_k$ singularities

TL;DR

This paper extends local intersection inequalities to weighted blowups and applies them to establish birational rigidity of certain Fano 3-folds with specific singularities, advancing understanding of their birational properties.

Contribution

It generalizes an intersection inequality to weighted blowups and derives a new inequality for $cA_k$ singularities, leading to results on birational rigidity.

Findings

Established the $4n^2/(k+1)$-inequality for $cA_k$ singularities.

Proved birational rigidity for families of Fano 3-folds with $cA_k$ singularities.

Extended intersection-theoretic inequalities to weighted blowups.

Abstract

We generalize an intersection-theoretic local inequality of Fulton-Lazarsfeld to weighted blowups. As a consequence, we obtain the $4n^2/(k+1)$-inequality for isolated $cA_k$ singularities, an analogue of the $4 n^2$-inequality for smooth points. We use this to prove birational rigidity of many families of Fano 3-fold weighted complete intersections with terminal quotient singularities and isolated $cA_k$ singularities, including sextic double solids with $cA_1$ and ordinary $cA_2$ points.

Tables

Table 1. Table 1 : Fano 3 3 -fold weighted hypersurfaces of index 1 1

№	$X_d\subset \mathbb{P}(a_0,\dots ,a_4)$	$-K_X^3$	$l_{\mathrm{ic}}$	$k_{\mathrm{cA}}$	Case	$F(0,0,z,t,w)$
6	$X_8\subset \mathbb{P}(1,1,1,2,4)$	$1$		$1$	$★$
10	$X_{10}\subset \mathbb{P}(1,1,1,3,5)$	$2/3$		$1$	$★$
11	$X_{10}\subset \mathbb{P}(1,1,2,2,5)$	$1/2$	$2$	$3$	$♣$
14	$X_{12}\subset \mathbb{P}(1,1,1,4,6)$	$1/2$		$1$	$★$
15	$X_{12}\subset \mathbb{P}(1,1,2,3,6)$	$1/3$		$1$	$★$
17	$X_{12}\subset \mathbb{P}(1,1,3,4,4)$	$1/4$	$4$	$3$	$♡$	$z^4+{\varphi }_{\left(17\right)}(t,w)$
18	$X_{12}\subset \mathbb{P}(1,2,2,3,5)$	$1/5$		$1$	$★$
19	$X_{12}\subset \mathbb{P}(1,2,3,3,4)$	$1/6$	$6$	$3$	$♡$	$w^3+{\varphi }_{\left(19\right)}(z,t)$
21	$X_{14}\subset \mathbb{P}(1,1,2,4,7)$	$1/4$	$4$	$3$	$♡$	$w^2+z{\varphi }_{\left(21\right)}(z,t)$
22	$X_{14}\subset \mathbb{P}(1,2,2,3,7)$	$1/6$	$6$	$3$	$♣$
25	$X_{15}\subset \mathbb{P}(1,1,3,4,7)$	$5/28$	$7$	$2$	$♡$	$\underset{¯}{w{t}^2}+\underset{¯}{t^{3}z}+z^5$
26	$X_{15}\subset \mathbb{P}(1,1,3,5,6)$	$1/6$	$6$	$3$	$♡$	$w^{2}z+t^3+z^5$
27	$X_{15}\subset \mathbb{P}(1,2,3,5,5)$	$1/10$	$10$	$3$	$♡$	${\varphi }_{\left(27\right)}(t,w)+z^5$
28	$X_{15}\subset \mathbb{P}(1,3,3,4,5)$	$1/12$	$12$	$3$	$♣$
29	$X_{16}\subset \mathbb{P}(1,1,2,5,8)$	$1/5$	$5$	$3$	$♡$	$w^2+\left(t^2{z}^3\right)+z^8$
30	$X_{16}\subset \mathbb{P}(1,1,3,4,8)$	$1/6$	$4$	$5$	$♡$	$w^2+t^4+\left(t{z}^4\right)$
31	$X_{16}\subset \mathbb{P}(1,1,4,5,6)$	$2/15$	$6$	$4$	$♡$	$w^{2}z+\left(w{t}^2\right)+z^4$
32	$X_{16}\subset \mathbb{P}(1,2,3,4,7)$	$2/21$	$14$	$2$	$♡$	$\left(w{z}^3\right)+t^4+\left(t{z}^4\right)$
34	$X_{18}\subset \mathbb{P}(1,1,2,6,9)$	$1/6$	$6$	$3$	$♡$	$w^2+c(z,t)$
35	$X_{18}\subset \mathbb{P}(1,1,3,5,9)$	$2/15$	$5$	$5$	$♡$	$w^2+t^{3}z+z^6$
36	$X_{18}\subset \mathbb{P}(1,1,4,6,7)$	$3/28$	$7$	$4$	$♡$	$w^{2}z+t^3+t{z}^3$
37	$X_{18}\subset \mathbb{P}(1,2,3,4,9)$	$1/12$	$6$	$7$	$♡$	$w^2+\left(t^3{z}^2\right)+z^6$
38	$X_{18}\subset \mathbb{P}(1,2,3,5,8)$	$3/40$	$10$	$4$	$♡$	$\underset{¯}{w{t}^2}+\underset{¯}{t^{3}z}+z^6$
41	$X_{20}\subset \mathbb{P}(1,1,4,5,10)$	$1/10$	$5$	$7$	$♡$	$w^2+t^4+z^5$
42	$X_{20}\subset \mathbb{P}(1,2,3,5,10)$	$1/15$	$10$	$5$	$♡$	$w^2+t^4+\left(t{z}^5\right)$
43	$X_{20}\subset \mathbb{P}(1,2,4,5,9)$	$1/18$	$18$	$3$	$♡$	$t^4+z^5$
44	$X_{20}\subset \mathbb{P}(1,2,5,6,7)$	$1/21$	$14$	$5$	$♡$	$w^{2}t+z^4$
45	$X_{20}\subset \mathbb{P}(1,3,4,5,8)$	$1/24$	$24$	$3$	$♡$	$w^{2}z+t^4+z^5$
46	$X_{21}\subset \mathbb{P}(1,1,3,7,10)$	$1/10$	$10$	$3$	$♡$	$t^3+z^7$
47	$X_{21}\subset \mathbb{P}(1,1,5,7,8)$	$3/40$	$8$	$5$	$♡$	$w^{2}z+t^3$
48	$X_{21}\subset \mathbb{P}(1,2,3,7,9)$	$1/18$	$18$	$3$	$♡$	$w^{2}z+t^3+z^7$
49	$X_{21}\subset \mathbb{P}(1,3,5,6,7)$	$1/30$	$15$	$7$	$♡$	$w^3+\left(t{z}^3\right)$
51	$X_{22}\subset \mathbb{P}(1,1,4,6,11)$	$1/12$	$6$	$7$	$♡$	$w^2+t^{3}z+z^{4}t$
52	$X_{22}\subset \mathbb{P}(1,2,4,5,11)$	$1/20$	$10$	$7$	$♠$	$w^2+\left(t^2{z}^3\right)$
53	$X_{24}\subset \mathbb{P}(1,1,3,8,12)$	$1/12$	$8$	$5$	$♡$	$w^2+t^3+z^8$
54	$X_{24}\subset \mathbb{P}(1,1,6,8,9)$	$1/18$	$9$	$7$	$♡$	$w^{2}z+t^3+z^4$
55	$X_{24}\subset \mathbb{P}(1,2,3,7,12)$	$1/21$	$14$	$5$	$♡$	$w^2+t^{3}z+z^8$
56	$X_{24}\subset \mathbb{P}(1,2,3,8,11)$	$1/22$	$22$	$3$	$♡$	$t^3+z^8$
57	$X_{24}\subset \mathbb{P}(1,3,4,5,12)$	$1/30$	$15$	$7$	$♡$	$w^2+t^{4}z+z^6$
58	$X_{24}\subset \mathbb{P}(1,3,4,7,10)$	$1/35$	$30$	$3$	$♡$	$w^{2}z+\left(w{t}^2\right)+z^6$
59	$X_{24}\subset \mathbb{P}(1,3,6,7,8)$	$1/42$	$21$	$7$	$♡$	$w^3+z^4$
60	$X_{24}\subset \mathbb{P}(1,4,5,6,9)$	$1/45$	$36$	$4$	$♡$	$w^{2}t+\left(w{z}^3\right)+t^4$
61	$X_{25}\subset \mathbb{P}(1,4,5,7,9)$	$5/252$	$36$	$4$	$♡$	$w^{2}t+z^5$
62	$X_{26}\subset \mathbb{P}(1,1,5,7,13)$	$2/35$	$7$	$9$	$♡$	$w^2+t^{3}z$
63	$X_{26}\subset \mathbb{P}(1,2,3,8,13)$	$1/24$	$8$	$11$	$♠$	$w^2+\left(t{z}^6\right)$
64	$X_{26}\subset \mathbb{P}(1,2,5,6,13)$	$1/30$	$10$	$11$	$♡$	$w^2+\left(t{z}^4\right)$
65	$X_{27}\subset \mathbb{P}(1,2,5,9,11)$	$3/110$	$22$	$5$	$♡$	$w^{2}z+t^3$
66	$X_{27}\subset \mathbb{P}(1,5,6,7,9)$	$1/70$	$35$	$7$	$♡$	$w^3+w{z}^3+t^{3}z$
67	$X_{28}\subset \mathbb{P}(1,1,4,9,14)$	$1/18$	$9$	$7$	$♡$	$w^2+z^7$
68	$X_{28}\subset \mathbb{P}(1,3,4,7,14)$	$1/42$	$21$	$7$	$♡$	$w^2+t^4+z^7$
69	$X_{28}\subset \mathbb{P}(1,4,6,7,11)$	$1/66$	$44$	$5$	$♡$	$w^{2}z+t^4$
70	$X_{30}\subset \mathbb{P}(1,1,4,10,15)$	$1/20$	$10$	$7$	$♡$	$w^2+t^3+t{z}^5$
71	$X_{30}\subset \mathbb{P}(1,1,6,8,15)$	$1/24$	$8$	$11$	$♡$	$w^2+t^{3}z+z^5$
72	$X_{30}\subset \mathbb{P}(1,2,3,10,15)$	$1/30$	$10$	$11$	$♡$	$w^2+t^3+z^{10}$
73	$X_{30}\subset \mathbb{P}(1,2,6,7,15)$	$1/42$	$14$	$11$	$♡$	$w^2+z^5$
74	$X_{30}\subset \mathbb{P}(1,3,4,10,13)$	$1/52$	$39$	$4$	$♡$	$w^{2}z+t^5+t{z}^5$
75	$X_{30}\subset \mathbb{P}(1,4,5,6,15)$	$1/60$	$20$	$11$	$♡$	$w^2+t^5+z^6$
76	$X_{30}\subset \mathbb{P}(1,5,6,8,11)$	$1/88$	$55$	$5$	$♡$	$w^{2}t+t^{3}z+z^5$
77	$X_{32}\subset \mathbb{P}(1,2,5,9,16)$	$1/45$	$18$	$9$	$♡$	$w^2+t^{3}z$
78	$X_{32}\subset \mathbb{P}(1,4,5,7,16)$	$1/70$	$28$	$9$	$♡$	$w^2+z^{5}t$
79	$X_{33}\subset \mathbb{P}(1,3,5,11,14)$	$1/70$	$42$	$5$	$♡$	$w^{2}z+t^3$
80	$X_{34}\subset \mathbb{P}(1,3,4,10,17)$	$1/60$	$30$	$7$	$♡$	$w^2+t^{3}z+z^{6}t$
81	$X_{34}\subset \mathbb{P}(1,4,6,7,17)$	$1/84$	$28$	$11$	$♡$	$w^2+t^{4}z$
82	$X_{36}\subset \mathbb{P}(1,1,5,12,18)$	$1/30$	$12$	$9$	$♡$	$w^2+t^3$
83	$X_{36}\subset \mathbb{P}(1,3,4,11,18)$	$1/66$	$33$	$7$	$♡$	$w^2+z^9$
84	$X_{36}\subset \mathbb{P}(1,7,8,9,12)$	$1/168$	$63$	$9$	$♡$	$w^3+w{z}^3+t^4$
85	$X_{38}\subset \mathbb{P}(1,3,5,11,19)$	$2/165$	$33$	$9$	$♡$	$w^2+t^{3}z$
86	$X_{38}\subset \mathbb{P}(1,5,6,8,19)$	$1/120$	$40$	$11$	$♡$	$w^2+t^{4}z+t{z}^5$
87	$X_{40}\subset \mathbb{P}(1,5,7,8,20)$	$1/140$	$40$	$13$	$♡$	$t^5+w^2$
88	$X_{42}\subset \mathbb{P}(1,1,6,14,21)$	$1/42$	$14$	$11$	$♡$	$z^7+t^3+w^2$
89	$X_{42}\subset \mathbb{P}(1,2,5,14,21)$	$1/70$	$14$	$19$	$♡$	$t^3+w^2$
90	$X_{42}\subset \mathbb{P}(1,3,4,14,21)$	$1/84$	$42$	$7$	$♡$	$z^{7}t+t^3+w^2$
91	$X_{44}\subset \mathbb{P}(1,4,5,13,22)$	$1/130$	$52$	$9$	$♡$	$z{t}^3+w^2$
92	$X_{48}\subset \mathbb{P}(1,3,5,16,24)$	$1/120$	$48$	$9$	$♡$	$t^3+w^2$
93	$X_{50}\subset \mathbb{P}(1,7,8,10,25)$	$1/280$	$70$	$15$	$♡$	$z^{5}t+t^5+w^2$
94	$X_{54}\subset \mathbb{P}(1,4,5,18,27)$	$1/180$	$36$	$19$	$♡$	$w^2+t^3$
95	$X_{66}\subset \mathbb{P}(1,5,6,22,33)$	$1/330$	$110$	$11$	$♡$	$w^2+t^3+z^{11}$

Table 2. Table 2 : Fano 3 3 -fold WCIs of codimension 2 2 and index 1 1

№	$X_{d_1,d_2}\subset \mathbb{P}(a_0,\dots ,a_5)$	$-K_X^3$	$l_{\mathrm{ic}}$	$k_{\mathrm{cA}}$	Case
8	$X_{4,6}\subset \mathbb{P}(1,1,2,2,2,3)$	$1$		$1$	$★$
14	$X_{6,6}\subset \mathbb{P}(1,2,2,2,3,3)$	$1/2$	$2$	$3$	${♣}_{4,5}$
20	$X_{6,8}\subset \mathbb{P}(1,2,2,3,3,4)$	$1/3$		$1$	$★$
24	$X_{6,10}\subset \mathbb{P}(1,2,2,3,4,5)$	$1/4$		$1$	$★$
31	$X_{8,10}\subset \mathbb{P}(1,2,3,4,4,5)$	$1/6$		$1$	$★$
37	$X_{8,12}\subset \mathbb{P}(1,2,3,4,5,6)$	$2/15$		$1$	$★$
45	$X_{10,12}\subset \mathbb{P}(1,2,4,5,5,6)$	$1/10$		$1$	$★$
47	$X_{10,12}\subset \mathbb{P}(1,3,4,4,5,6)$	$1/12$	$12$	$3$	${♣}_{4,5}$
51	$X_{10,14}\subset \mathbb{P}(1,2,4,5,6,7)$	$1/12$	$12$	$3$	${♣}_{3,5}$
59	$X_{12,14}\subset \mathbb{P}(1,4,4,5,6,7)$	$1/20$	$20$	$3$	${♣}_{4,5}$
60	$X_{12,14}\subset \mathbb{P}(2,3,4,5,6,7)$	$1/30$	$20$	$5$	${♣}_{4,5}$
64	$X_{12,16}\subset \mathbb{P}(1,2,5,6,7,8)$	$2/35$		$1$	$★$
71	$X_{14,16}\subset \mathbb{P}(1,4,5,6,7,8)$	$1/30$	$30$	$3$	${♡}_{4,5}$
75	$X_{14,18}\subset \mathbb{P}(1,2,6,7,8,9)$	$1/24$	$24$	$3$	${♡}_{3,5}$
76	$X_{12,20}\subset \mathbb{P}(1,4,5,6,7,10)$	$1/35$	$35$	$3$	${♡}_{3,5}$
78	$X_{16,18}\subset \mathbb{P}(1,4,6,7,8,9)$	$1/42$	$42$	$3$	${♡}_{4,5}$
84	$X_{18,30}\subset \mathbb{P}(1,6,8,9,10,15)$	$1/120$	$40$	$11$	${♡}_{3,5}$
85	$X_{24,30}\subset \mathbb{P}(1,8,9,10,12,15)$	$1/180$	$90$	$7$	${♡}_{4,5}$

Table 3. Table 3 : Equations for codimension 2 2 WCIs belonging to families marked ♡ \heartsuit

№	$F_1(0,0,z,t,v,w)$	$F_2(0,0,z,t,v,w)$	Case	$\Xi$	Remark
$71$	$wt+v^2$	$w^2+t{z}^2$	$t{z}^2\in F_2$	$\{P_z,P_t\}$
	$wt+v^2$	$w^2$	$t{z}^2\notin F_2$	$\left(x=y=v=w=0\right)$	$z^{3}x\in F_2$
$75$	$vz+t^2$	$w^2+z^3$		$\left\{P_v\right\}$
$76$	$vz+t^2$	$w^2+v^{2}t+z^4$		$\left\{P_v\right\}$
$78$	$wt+v^2$	$w^2+z^3$		$\left\{P_t\right\}$
$84$	$vz+t^2$	$w^2+v^3$		$\left\{P_z\right\}$
$85$	$wz+v^2$	$w^2+v{z}^2+t^3$		$\left\{P_z\right\}$

Equations

$${\mathrm{mult}}_P(D_1\cdot D_2)>4n^2.$$

$${\mathrm{mult}}_P(D_1\cdot D_2)>\frac{4}{k+1}{n}^2.$$

$${\mathrm{mult}}_P(D_1\cdot \dots \cdot D_d)$$

$${\mathrm{deg}}_E({\tilde{D}}_1\cdot \dots \cdot {\tilde{D}}_d)$$

$${\mathrm{mult}}_P(D_1\cdot \dots \cdot D_d)$$

$${\mathrm{deg}}_E({\tilde{D}}_1\cdot \dots \cdot {\tilde{D}}_d)$$

$$R_Q=\{\begin{array}{rlrl}& {\mathcal{O}}_{X,P}\left[\left\{t^j{x}_i|i\in \{1,\dots ,d\},j\in \{0,\dots ,w_i\}\right\}\right]& & \text{if Q=P},\\ & {\mathcal{O}}_{X,Q}[t]& & \text{otherwise},\end{array}$$

$$\mathbf{w}(g):=\inf \left\{\sum w_i{a}_i|\text{the monomial x1a1⋅…⋅xdad has nonzero coefficient in g}\right\},$$

$$\mathbf{w}(f)=\sup \{j\in {\mathbb{Z}}_{\ge 0}\mid t^{j}f\in R_P\}.$$

$$\mathbf{w}(f)\ge k⟺t^{k}f\in R_P$$

$$\frac{{\mathcal{O}}_{X,P}}{(x_1,\dots ,x_d{)}^{k+1}}\to \frac{\mathbb{k}[[x_1,\dots ,x_d]]}{(x_1,\dots ,x_d{)}^{k+1}}.$$

$${\left\{t^j{\bar{x}}_i\right\}}_{i,j}:=\left\{t^j{\bar{x}}_i\mid i\in \{1,\dots ,d\},j\in \{0,\dots ,w_i\}\right\}.$$

$$\frac{{\mathcal{O}}_X(X)}{(f)}\left[{\left\{t^j{\bar{x}}_i\right\}}_{i,j}\right]=\frac{S}{(f,tf,\dots ,t^{\mathbf{w}(f)}f)}.$$

$$\begin{array}{rl}{Z}_i(X{)}_{\mathbb{Q}}& :=Z_i(X){\otimes }_{\mathbb{Z}}\mathbb{Q},\\ A_i(X{)}_{\mathbb{Q}}& :=A_i(X){\otimes }_{\mathbb{Z}}\mathbb{Q}.\end{array}$$

$$\mathrm{deg}:A_0(X{)}_{\mathbb{Q}}\to \mathbb{Q}.$$

$${\mathrm{mult}}_S(\alpha ):=\sum a_i{\mathrm{mult}}_S(Z_i),$$

$$(b_{1}D_{1})\cdot\ldots\cdot(b_{s}D_{s})\in A_{d-s}\bigl{(}\operatorname{Supp}(D_{1})\cap\ldots\cap\operatorname{Supp}(D_{s})\bigr{)}.$$

$$D_{1}\cdot\ldots\cdot D_{s}:=\frac{(b_{1}D_{1})\cdot\ldots\cdot(b_{d}D_{d})}{b_{1}\cdots b_{d}}\in A_{d-s}\bigl{(}\operatorname{Supp}(D_{1})\cap\ldots\cap\operatorname{Supp}(D_{s})\bigr{)}_{\mathbb{Q}}.$$

$${\mathrm{mult}}_P(Z)={\mathrm{mult}}_P(Z\cdot H_1\cdot \dots \cdot H_i).$$

$${\mathrm{mult}}_P(Z)={\mathrm{mult}}_P(Z\cdot H_1\cdot \dots \cdot H_j).$$

$${\mathrm{mult}}_P(Z\cdot H_1\cdot \dots \cdot H_i)={\mathrm{mult}}_P(Z\cdot H_1\cdot \dots \cdot H_j),$$

$${\mathrm{mult}}_P(D_1\cdot \dots \cdot D_d)={\mathrm{mult}}_0\left(V(f_1^{\le N})\cdot \dots \cdot V(f_d^{\le N})\right).$$

$${\mathrm{mult}}_P(D_1\cdot \dots \cdot D_d)=l_{{\mathcal{O}}_{X,P}}({\mathcal{O}}_{X,P}/(F_1,\dots ,F_d)).$$

$${\mathrm{mult}}_P(D_1\cdot \dots \cdot D_d)=l_{\mathbb{k}[[x_1,\dots ,x_d]]}\left(\frac{\mathbb{k}[[x_1,\dots ,x_d]]}{(f_1,\dots ,f_d)}\right).$$

$${\mathrm{mult}}_P(D_1\cdot \dots \cdot D_d)={\dim }_{\mathbb{k}}\frac{\mathbb{k}[[x_1,\dots ,x_d]]}{(f_1,\dots ,f_d)}.$$

$${\dim }_{\mathbb{k}}\frac{\mathbb{k}[[x_1,\dots ,x_d]]}{(f_1,\dots ,f_d)}={\dim }_{\mathbb{k}}\frac{\mathbb{k}[[x_1,\dots ,x_d]]}{(f_1^{\le N},\dots ,f_d^{\le N})}.$$

$${\dim }_{\mathbb{k}}\frac{\mathbb{k}[[x_1,\dots ,x_d]]}{(f_1^{\le N},\dots ,f_d^{\le N})}={\mathrm{mult}}_0\left(V(f_1^{\le N})\cdot \dots \cdot V(f_d^{\le N})\right).\text{qed}$$

$${\mathrm{ord}}_P(D):=\sup \{j\ge 0\mid f\in {\mathfrak{m}}_{X,P}^j\},$$

$${\mathrm{ord}}_P(D):=\sum a_i{\mathrm{ord}}_P(D_i),$$

$$\Gamma :=\mathrm{Supp}(S_1)\cap \cdots \cap \mathrm{Supp}(S_{d-3})\cap \mathrm{Supp}(D_1)\cap \mathrm{Supp}(D_2)$$

$$T\cdot S_1\cdot \dots \cdot S_{d-3}\cdot D_1\cdot D_2\ge {\mathrm{ord}}_P(T){\mathrm{ord}}_P(S_1)\cdots {\mathrm{ord}}_P(S_{d-3}){\mathrm{mult}}_P(D_1\cdot D_2).$$

Full text

1. 1 Introduction

2. 1.1 Approach for local inequalities for $cA_{k}$-points

3. 1.2 Sextic double solids

4. 1.3 Rigidity of singular Fano varieties

5. 1.4 Other works on birational rigidity of singular varieties

6. 2 Preliminaries

7. 2.1 Weighted blowups

8. 2.2 Intersection theory

9. 2.3 The method of maximal singularities for Fano 3-folds

10. 2.4 Divisorial contractions to $cA_{k}$-points

11. 3 Local inequalities

12. 3.1 Main inequality

13. 3.2 Intersection cycles of mobile linear systems around LCI singularities

14. 3.3 Local inequality for $cA_{k}$-singularities

15. 3.4 Algebraic corollary

16. 4 Generalizations to (LCI-)quotient singularities

17. 4.1 Main inequality for cyclic quotient singularities

18. 4.2 Intersection cycles of mobile linear systems around LCI-quotient singularities

19. 5 Sextic double solids

20. 5.1 Birational involutions

21. 5.2 Birational rigidity of sextic double solids

22. 6 Fano 3-fold WCIs of index 1

23. 6.1 Basics on weighted projective varieties

24. 6.2 Proof of Theorem 6.2

25. 6.2.1 Weighted hypersurfaces

26. 6.2.2 Weighted complete intersections of codimension 2

27. 6.2.3 Completion of Proof of Theorem 6.2

Introduction

The notion of birational rigidity was originally introduced in the context of proving non-rationality of Fano varieties. It is also crucial for the classification of Fano varieties up to birational equivalence.

Definition 1.1**.**

We say that a Fano variety is birationally rigid if it is the unique Mori fiber space in its birational class. We say that a birationally rigid Fano variety $X$ is superrigid if $\operatorname{Aut}(X)=\operatorname{Bir}(X)$.

Birational rigidity is typically proven using the Noether–Fano method: non-rigidity of a Fano variety $X$ implies the existence of a mobile linear system $\mathcal{M}\subset\left|-nK_{X}\right|$ such that the pair $(X,\frac{1}{n}\mathcal{M})$ has worse than canonical singularities. Thus, interpreting non-canonicity of the pair $(X,\frac{1}{n}\mathcal{M})$ in geometric terms is the key to being able to prove the birational rigidity of a Fano variety. The $4n^{2}$-inequality is one of the most important ingredients for proof of birational rigidity of Fano varieties.

Theorem 1.2** ($4n^{2}$-inequality, [Puk13, Theorem 2.1]).**

Let $P\in X$ be the germ of a smooth $3$-fold. Let $\mathcal{M}$ be a mobile linear system on $X$ and let $n>0$ be a rational number. If $P$ is a center of non-canonical singularities of the pair $(X,\frac{1}{n}\mathcal{M})$, then for general members $D_{1},D_{2}$ in $\mathcal{M}$ we have

[TABLE]

Quasismooth Fano weighted hypersurfaces of index $1$ are proven to be birationally rigid using the $4n^{2}$-inequality and the uniqueness of divisorial contractions with center a cyclic quotient singularity ([CP17], [CPR00]).

In this paper we prove an analogue of Theorem 1.2 for compound Du Val singularities of type $cA_{k}$.

Theorem 1.3** (= Theorem 3.8).**

Let $P\in X$ be an isolated $cA_{k}$ singularity, where $k\geq 1$, let $\mathcal{M}$ be a mobile linear system of Cartier divisors and $n>0$ a rational number. Suppose that $P$ is a non-canonical center of the pair $(X,\frac{1}{n}\mathcal{M})$. Then for general members $D_{1},D_{2}\in\mathcal{M}$, we have

[TABLE]

As an application, we consider the birational rigidity of sextic double solids and also other weighted complete intersection index $1$ Fano varieties.

Theorem 1.4** ($=$ Theorem 6.2).**

Let $X$ be a prime Fano $3$-fold weighted complete intersection which belongs to one of the families listed in Tables LABEL:table:Fanohyp and 2. Suppose that $X$ is quasismooth along the singular locus of the ambient weighted projective space, and $X$ has only $cA_{k}$ singularities besides terminal quotient singular points. Suppose further that $k$ is not greater than the bound $k_{\mathrm{cA}}$ listed in the table. Then $X$ is birationally rigid.

Approach for local inequalities for $cA_{k}$-points

Theorem 1.3 is a generalization of [KOPP24, Theorem 1.2] but we achieve it with different methods. In [KOPP24] we considered a tower of blowups over $P\in X$ and we tracked the intersections on every level of the tower. The key ingredient for this approach is the following result.

Theorem 1.5** ([FL82, Theorem 3]).**

Let $P$ be a $\Bbbk$-rational point of a $d$-dimensional smooth integral algebraic $\Bbbk$-scheme $X$, where $\Bbbk$ is a field and $d\geq 2$. Let $\varphi\colon Y\to X$ be the blow up at $P$ and let $E$ be the exceptional prime divisor. Let $D_{1},\ldots,D_{d}$ be effective $\mathbb{Q}$-divisors on $X$ such that $P$ is an isolated point of $D_{1}\cap\ldots\cap D_{d}$. For every $i$, let $\tilde{D}_{i}$ be the strict transform of $D_{i}$. Then, both of the following hold:

[TABLE]

where $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ denotes the degree of the [math]-cycle class $\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d}$ with coefficients in $\mathbb{Q}$ on a small neighborhood of $E$. Moreover, $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ is zero if and only if $\tilde{D}_{1}\cap\ldots\cap\tilde{D}_{d}\cap E$ is empty.

Given a variety $X$ of dimension $3$ with a $cA_{1}$ singularity $P$ embedded in a $4$-dimensional variety $Y$ such that $(P\in Y)$ is smooth and two Cartier divisors $D_{1},D_{2}$ on $X$, we can apply this theorem to $X,D_{1}^{\prime},D_{2}^{\prime},H$, where $D_{i}^{\prime}$ is some extension of $D_{i}$ to $Y$ and $H$ is a general very ample divisor passing through $P$. This provides a way to estimate intersection multiplicities using multiplicities of divisors at points which allowed us to prove [KOPP24, Theorem 1.2].

It was possible since on every level we had a factorial variety. For $cA_{2}$-singularities, the exceptional divisor of the blow up is the sum of two non-Cartier divisors, and if we want to use the similar approach, then we have to consider factorialization of every level of the tower of blow ups. In this case the computations become very cumbersome, so we opted for a different approach.

By Theorem 2.16 ([Kaw03, Theorem 1.13], [Kaw02, Theorem 1.1] and [Yam18, Theorem 2.6]), we know that every divisorial contraction to a $cA_{k}$ point is a weighted blowup with weights from a specific list. This allows us to consider intersections for a single weighted blowup instead of tracking intersections through a tower of blowups. On the other hand, in order to make this work we have to generalize [FL82, Theorem 3] to weighted blowups.

Theorem 1.6** (=Theorem 3.1).**

Let $P$ be a closed point of a $d$-dimensional regular integral algebraic $\Bbbk$-scheme $X$, where $\Bbbk$ is a field and $d\geq 2$. Let ${\bm{w}}=(w_{1},\ldots,w_{d})$ be a tuple of positive integers such that $\gcd{\bm{w}}=1$. Let $\varphi\colon Y\to X$ be the ${\bm{w}}$-blowup with respect to a regular system of parameters at $P$ and let $E$ be the exceptional prime divisor. Let $D_{1},\ldots,D_{d}$ be effective $\mathbb{Q}$-divisors on $X$ such that $P$ is an isolated point of $\operatorname{Supp}D_{1}\cap\ldots\cap\operatorname{Supp}D_{d}$. For every $i$, let $\tilde{D}_{i}$ be the strict transform of $D_{i}$. Then, both of the following hold:

[TABLE]

where $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ denotes the degree of the [math]-cycle class $\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d}$ with coefficients in $\mathbb{Q}$ on a small neighborhood of $E$ and $[\kappa(P):\Bbbk]$ is the degree of the field extension. Moreover, $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ is zero if and only if $\operatorname{Supp}\tilde{D}_{1}\cap\ldots\cap\operatorname{Supp}\tilde{D}_{d}\cap E$ is empty.

Sextic double solids

We previously conjectured that factorial sextic double solids with $cA_{2}$-singularities are birationally superrigid ([KOPP24, Conjecture 1.3]). In this paper we consider factorial sextic double solids with ordinary $cA_{2}$-points (see Definition 2.19) and it turns out that they are birationally rigid but might not be superrigid. There are two possible divisorial contractions centered on an ordinary $cA_{2}$ point: regular and exceptional. The regular one generates an involution in general (Lemma 5.5) and in special cases generates a “bad link” (Lemma 5.4). Using Theorem 1.3 we show that the exceptional one does not generate a Sarkisov link. It follows that any birational map $X\dashrightarrow Y$ from a sextic double solid $X$ with $cA_{2}$-singularities to a Mori fiber space $Y$ is decomposed into involutions of $X$ and a map defined at $cA_{2}$-singularities (see section 5). This results in the following theorem.

Theorem 1.7** ($=$ Theorem 5.7).**

Let $X$ be a factorial sextic double solid with only isolated $cA_{1}$ points and ordinary $cA_{2}$ points. Then $X$ is birationally rigid.

Sextic double solids with non-ordinary $cA_{2}$-singularities remain out of reach since they admit divisorial contractions with weights $(3,3,2,1)$ and possibly more. Theorem 1.3 is not enough to exclude these contractions and there are infinitely many of them by [Pae24b] which makes it unrealistic to consider all potential links originating at $cA_{2}$-singularities.

Rigidity of singular Fano varieties

According to Noether–Fano inequality, existence of a birational map $X\dashrightarrow Y$ to another Mori fiber space implies existence of $n$ and a mobile linear subsystem $\mathcal{M}\subset\left|-nK_{X}\right|$ such that the pair $(X,\frac{1}{n}\mathcal{M})$ has worse than canonical singularities at some subvariety $Z\subset X$. The subvariety $Z$ may be a curve, a non-singular point, or a quotient singularity, or a $cA_{k}$-point. Thanks to the previous results: [CPR00], [CP17], [Oka14], [AZ16], and [KOPP24] we only need to consider $cA_{k}$-points.

For each $cA_{k}$-singular point $P$ we produce a linear system $\mathcal{H}$ such that $P$ is an isolated component of the base locus, we say that the linear equivalence class of $\mathcal{H}$ is a $P$-isolating class. Let $D_{1},D_{2}$ be general divisors in $\mathcal{M}$ and let $H$ be a general divisor in $\mathcal{H}$, then the intersection $D_{1}\cdot D_{2}\cdot H$ provides a natural upper bound on $\operatorname{mult}_{P}(D_{1}\cap D_{2})$. In order to prove birational rigidity we present a $P$-isolating class of sufficiently low degree to get a contradiction with Theorem 1.3 for every family in the tables LABEL:table:Fanohyp and 2.

It is interesting to note that we proved for several number of families of index $1$ Fano complete intersections that $\mathbb{Q}$-factorial members with $cA_{11}$-points are birationally rigid, and there is even family №89 such that $\mathbb{Q}$-factorial members with $cA_{19}$-point is birationally rigid. But we are not sure if there exist $\mathbb{Q}$-factorial members with such singularities. It was shown in ([Pae24a, Theorem A(e)]) that general sextic double solids with an isolated $cA_{k}$-point are $\mathbb{Q}$-factorial for $k=1,\dots,6,8$ and the space of sextic double solids with an isolated $cA_{7}$-point has four connected components, one of which has no $\mathbb{Q}$-factorial members. Thus we expect that for most families general member with a $cA_{k}$-point is $\mathbb{Q}$-factorial for a small $k$, but we do not know about big $k$. This motivates the following question.

Question 1.8**.**

For Fano weighted complete intersections, what is the highest $k$ such that the general member of the family with a $cA_{k}$-point is $\mathbb{Q}$-factorial?

Other works on birational rigidity of singular varieties

Our inequality works very well for varieties of low degree, but birational rigidity results for singular varieties can be extended to higher degree Fano varieties as well. For example the birational rigidity of quartic threefolds with one $cA_{1}$-point is proven in [Puk24]. We expect that with a lot of extra effort similar results can be proven for the families that are not in tables LABEL:table:Fanohyp and 2.

Also note, that in higher dimensions one can obtain stronger inequalities, for example see [Puk17]. This approach has been applied to prove birational rigidity of many families of high dimensional varieties: hypersurfaces, cyclic covers, complete intersections ([Puk19a], [Puk19b], [EP18], [EP19]).

Acknowledgments

The first author was supported by IBS-R003-D1 grant. The second author was supported by JAPS KAKENHI Grant Numbers 23K22389 and 24K00519. The third author was supported by SFB 195 No. 286237555 of DFG, the Simons Foundation, grant SFI-MPS-T-Institutes-00007697, and the Ministry of Education and Science of the Republic of Bulgaria, grant DO1-239/10.12.2024.

Preliminaries

An algebraic $\Bbbk$-scheme, where $\Bbbk$ is a field, is a scheme of finite type over $\Bbbk$. A variety is an integral separated scheme of finite type over the field of complex numbers $\mathbb{C}$. By the germ of a scheme $X$ at a point $P$, we mean the direct limit of open subschemes containing $P$.

Weighted blowups

In Definition 2.1, we define weighted blowups of arbitrary schemes $X$ (not necessarily of finite type over a field) along a regular system of parameters. This enables us to state Theorem 3.1 over an arbitrary field. Definition 2.1 coincides with the weighted blowup in [Kaw24, Definition 2.2.11] when $X$ is an algebraic variety (over $\mathbb{C}$). Definition 2.1 coincides with the (usual) blowup along $P$ if all the weights are $1$.

Definition 2.1**.**

Let $d$ be a positive integer. Let ${\bm{w}}=(w_{1},\ldots,w_{d})$ be a $d$-tuple of positive integers such that $\gcd{\bm{w}}:=\gcd(w_{1},\ldots,w_{d})=1$. Let $X$ be a scheme and $P\in X$ a regular closed point. We say that a morphism of schemes $\varphi\colon Y\to X$ is the weighted blowup with weights $w_{1},\ldots,w_{d}$ (or ${\bm{w}}$-blowup) with respect to a regular system of parameters $x_{1},\ldots,x_{d}$ at $P$ if $Y$ is isomorphic over $X$ to $\operatorname{Proj}_{X}R$, where $R$ is the $\mathbb{Z}_{\geq 0}$-graded $\mathcal{O}_{X}$-algebra with stalks

[TABLE]

where $t$ denotes the grading. The exceptional divisor of $\varphi$ is defined to be the subscheme of $Y$ given by the surjective $\mathcal{O}_{X}$-algebra homomorphism $R\to R/\bigoplus_{m\in\mathbb{Z}_{\geq 0}}R_{m+1}t^{m}$, where $R_{m}$ denotes the quasi-coherent $\mathcal{O}_{X}$-ideal sheaf that is the $m$-th graded piece of $R$.

Note that the exceptional divisor of a weighted blowup need not be Cartier. Next, we define the weight and the least weight part of an element of $\mathcal{O}_{X,P}$.

Definition 2.2**.**

Given a positive integer $d$ and vector $\bm{w}=(w_{1},\ldots,w_{d})$ of positive integers, the weight of an element $g$ of the power series ring $\Bbbk[[x_{1},\ldots,x_{d}]]$ is defined to be

[TABLE]

where the sum is over $i\in\{1,\ldots,d\}$ and the infimum is over all the monomials in $\Bbbk[x_{1},\ldots,x_{d}]$. The least weight part of $g\in\Bbbk[[x_{1},\ldots,x_{d}]]$, denoted $g^{\bm{w}}$, is defined to be the sum of all the monomials of weight $\bm{w}(g)$, together with their coefficients in $g$.

In the setting of Definition 2.1, the inclusion $\iota\colon\Bbbk[x_{1},\ldots,x_{d}]\to\mathcal{O}_{X,P}$ induces a $\Bbbk$-algebra isomorphism on the completions $\hat{\iota}\colon\Bbbk[[x_{1},\ldots,x_{d}]]\to\hat{\mathcal{O}}_{X,P}$. Define the local monomorphism $\chi\colon\mathcal{O}_{X,P}\to\Bbbk[[x_{1},\dots,x_{d}]]$ to be the composition of the natural injection $\mathcal{O}_{X,P}\to\hat{\mathcal{O}}_{X,P}$ and $\hat{\iota}^{-1}$. For every $f\in\mathcal{O}_{X,P}$, we define the weight of $f$ by $\bm{w}(f):=\bm{w}(\chi(f))$ and the least weight part of $f$ by $f^{\bm{w}}:=\iota(\chi(f)^{\bm{w}})$.

Lemma 2.3**.**

In the setting of Definition 2.1, for every $f\in\mathcal{O}_{X,P}$, we have

[TABLE]

Proof.

It suffices to prove the equivalence

[TABLE]

for every $f\in\mathcal{O}_{X,P}$ and positive integer $k$. By [Eis95, Theorem 7.1], $\chi$ induces a $\Bbbk$-algebra isomorphism

[TABLE]

So, it suffices to prove equivalence (2.1) in the case $f$ is a polynomial of degree at most $k$, where the statement is obvious. ∎

Below, if $X$ is affine, then $V_{X}(I)$ denotes the subscheme of $X$ defined by the ideal $I\subseteq\mathcal{O}_{X}(X)$ and $V_{Y}(J)$ denotes the subscheme of $Y$ defined by the homogeneous ideal $J\in R(X)$.

Lemma 2.4**.**

Let $d$ be a positive integer. Let ${\bm{w}}=(w_{1},\ldots,w_{d})$ be positive integers such that $\gcd{\bm{w}}=1$. Let $X$ be a scheme and $P$ a regular closed point. Let $\varphi\colon Y\to X$ be the ${\bm{w}}$-blowup with respect to a regular system of parameters $x_{1},\ldots,x_{d}$ at $P$. Then, all of the following hold:

1. (a)

if $X$ is an algebraic variety (over the field of complex numbers $\mathbb{C}$), then $\varphi$ coincides with the weighted blowup in **[Kaw24, Definition 2.2.11]** with $r=1$, 2. (b)

if $w_{1}=\ldots=w_{d}=1$, then $\varphi$ is the (usual) blowup along $P$, 3. (c)

$\varphi$* is the blowup along a quasi-coherent ideal sheaf,* 4. (d)

the exceptional divisor $E$ of $\varphi$ is isomorphic to the scheme $\mathbb{P}_{\kappa(P)}(w_{1},\ldots,w_{d}):=\operatorname{Proj}\kappa(P)[t^{w_{1}}x_{1},\ldots,t^{w_{d}}x_{d}]$ where $t$ denotes the grading and where $\kappa(P)$ is the residue field at $P$, 5. (e)

if $d\geq 2$ and $X$ is Noetherian and integral, then $E$ coincides with the exceptional locus of $\varphi$ with the induced reduced structure, 6. (f)

the local ring of $Y$ at $E$ is a discrete valuation ring and the corresponding valuation $v_{E}$ satisfies $v_{E}(f)=\bm{w}(f)$ for all $f\in\mathcal{O}_{X,P}$, 7. (g)

if $X$ is normal and integral, then so is $Y$, and 8. (h)

if $X$ is affine, then for all $f\in\mathcal{O}_{X}(X)\setminus\{0\}$, the strict transform of $V_{X}(f)$ is $V_{Y}(t^{\bm{w}(f)}f)$.

Proof.

We replace $X$ by an affine open neighborhood of $P$ where $x_{1},\ldots,x_{d}$ are defined. Denote $S:=R(X)$ and $S_{m}:=R_{m}(X)$. Let $S^{(n)}$ denote the $n$-th Veronese $\mathcal{O}_{X}(X)$-subalgebra of $S$, meaning $S^{(n)}:=\bigoplus_{m\in\mathbb{Z}_{\geq 0}}t^{m}S_{mn}$ where $t$ denotes the degree.

(a) Follows from the text below [Kaw24, Definition 2.2.13].

(b) Since $S$ is generated in degree $1$, $\varphi$ coincides with $\operatorname{Proj}\bigoplus_{m\in\mathbb{Z}_{\geq 0}}t^{m}S_{1}^{m}\to X$. The ideal $S_{1}$ is precisely the prime ideal defining $P$. Therefore, $\varphi$ is the blowup along $P$.

(c) For a divisible enough positive integer $n$, for example $n=d\cdot\operatorname{lcm}(w_{1},\ldots,w_{d})$, $S^{(n)}$ is generated in degree $1$ (see the proof of [QR22, Proposition 1.6.3] for details). So, $\varphi$ is the blowup along $S_{n}$.

(d) Follows from the fact that $E$ is given by $V_{Y}(\bigoplus_{m\in\mathbb{Z}_{\geq 0}}t^{m}S_{m+1})$.

(e) By (c), $\varphi$ is the blowup along $S_{n}$. For all positive integers $k<l$ and elements $f\in S_{l}$, we have $f^{l-k}\cdot(t^{l}f)^{k}=(t^{k}f)^{l}$. Therefore, the radical of $S_{n}$ is $\bigoplus_{m\in\mathbb{Z}_{\geq 0}}t^{m}S_{m+1}$. So, the support of the exceptional Cartier divisor of the blowup along $S_{n}$ coincides with $E$. Therefore, $\varphi$ restricts to an isomorphism $Y\setminus E\to X\setminus\{P\}$. By [GW20, Corollary 13.97], $Y$ is integral and $\varphi$ is proper and birational. Since $d\geq 2$, we can use [Liu06, Lemma 2.20] to conclude that the exceptional locus of $\varphi$ is $E$.

(f) Taking the affine open $U:=D_{+}(t^{w_{1}}x_{1})$ of $Y$, let $I\subseteq\mathcal{O}_{Y}(U)$ be the ideal defining $E\cap U$. Then, $\mathcal{O}_{Y}(U)$ consists of the elements $g/x_{1}^{k}$ where the nonnegative integer $k$ and element $g\in\mathcal{O}_{X}(X)$ are such that there exists a nonnegative integer $m$ satisfying $g\in S_{m}$ and $m\geq kw_{1}$. By Lemma 2.3, the coordinate ring $\mathcal{O}_{Y}(U)$, respectively the ideal $I$, consists of the elements $g/x_{1}^{k}$ such that $\bm{w}(g)\geq kw_{1}$, respectively $\bm{w}(g)>kw_{1}$. We see that $I$ is a prime ideal and the localization at the complement of $I$ is a discrete valuation ring. Since $\gcd(w_{1},\ldots,w_{d})=1$, the claim follows.

(g) We show that $Y$ is normal. By (f), for every nonnegative integer $m$, the ideal $S_{m}$ consists precisely of the elements $f\in\mathcal{O}_{X}(X)$ such that $v_{E}(f)\geq m$. It follows that $S_{m}$ is a normal ideal in $\mathcal{O}_{X}(X)$ for every nonnegative integer $m$, meaning that all the positive powers of $S_{m}$ are integrally closed in $\mathcal{O}_{X}(X)$. By (c), $\varphi$ is the blowup along $S_{n}$. Since $\mathcal{O}_{X}(X)$ is a normal domain, we find by [HS06, Proposition 5.2.4] that $\mathcal{O}_{X}(X)[tS_{n}]$ is a normal domain. Now, suffices to use the result that if $A$ is a $\mathbb{Z}_{\geq 0}$-graded normal domain, then $\operatorname{Proj}A$ is normal.

(h) By [QR22, Corollary 3.2.14], the strict transform $\overline{\varphi^{-1}(V_{X}(f))\setminus E}$ of $V_{X}(f)$ is given by the surjective $\mathbb{Z}_{\geq 0}$-graded $\mathcal{O}_{X}(X)$-algebra homomorphism $S\to\operatorname{Proj}\left(\frac{\mathcal{O}_{X}(X)}{(f)}\left[\left\{t^{j}\bar{x}_{i}\right\}_{i,j}\right]\right)$, where $\bar{x}_{i}$ denotes the equivalence class $x_{i}+(f)$ and where we use the notation

[TABLE]

Considering each homogeneous part separately, we find

[TABLE]

Since the intersections $(f,tf,\ldots,t^{\bm{w}(f)}f)\cap S_{\geq\bm{w}(f)}$ and $(t^{\bm{w}(f)}f)\cap S_{\geq\bm{w}(f)}$ coincide, the ideals $(f,tf,\ldots,t^{\bm{w}(f)}f)$ and $(t^{\bm{w}(f)}f)$ define the same subscheme of $Y$. The claim follows. ∎

Intersection theory

The degree of a $\mathbb{Z}$-cycle class is defined in [Ful98, Definition 1.4], the intersection of Cartier divisors is defined in [Ful98, Definition 2.4.2] and the intersection multiplicity of Cartier divisors at a point is defined in [Ful98, Example 7.1.10(a)]. In Definition 2.5 and Remark 2.7, we define the degree of a $\mathbb{Q}$-cycle class, the intersection of $\mathbb{Q}$-Cartier divisors and the intersection multiplicity of $\mathbb{Q}$-Cartier divisors at a point. In this section, the ground field $\Bbbk$ is an arbitrary field.

Definition 2.5**.**

Let $X$ be an algebraic $\Bbbk$-scheme and let $i$ be a positive integer. We denote by $Z_{i}(X)$ (resp. $A_{i}(X)$) the group of $i$-cycles on $X$ (resp. the group of $i$-cycles modulo rational equivalence on $X$). We define

[TABLE]

If $X$ is proper, then the degree map $\deg\colon A_{0}(X)\to\mathbb{Z}$ for [math]-cycles is extended to

[TABLE]

Definition 2.6**.**

Let $X$ be an algebraic $\Bbbk$-scheme and let $S\subset X$ be an integral subscheme. The multiplicity of $X$ along $S$, denoted by $\operatorname{mult}_{S}X$, is defined to be the Hilbert–Samuel multiplicity $e(\mathfrak{m}_{X,S},\mathcal{O}_{X,S})$ of the local ring $\mathcal{O}_{X,S}$ at its maximal ideal $\mathfrak{m}_{X,S}$. Let $Y$ be an algebraic $\Bbbk$-scheme, let $\alpha=\sum a_{i}Z_{i}$ be a rational cycle on $Y$, where $Z_{i}$ is an integral subscheme for any $i$ and $a_{i}\in\mathbb{Q}$, and let $S\subset Y$ be an integral subscheme of $Y$. Then the multiplicity of $\alpha$ along $S$ is defined by

[TABLE]

where we set $\operatorname{mult}_{S}(Z_{i})=0$ if $S$ is not contained in $Z_{i}$.

Remark 2.7*.*

Let $X$ be a $d$-dimensional normal integral algebraic $\Bbbk$-scheme. Let $D_{1},\dots,D_{s}$ be $\mathbb{Q}$-Cartier Weil divisors on $X$, where $2\leq s\leq d$. Let $b_{i}$ be positive integers such that $b_{i}D_{i}$ is a Cartier divisor. Then there is a well-defined $(d-s)$-cycle class

[TABLE]

We define

[TABLE]

Suppose in addition that $D_{1},\dots,D_{s}$ are effective. If $Z$ is a $(d-s)$-dimensional irreducible component of $\operatorname{Supp}(D_{1})\cap\cdots\cap\operatorname{Supp}(D_{s})$, then the coefficient of $[Z]$ in $D_{1}\cdot\ldots\cdot D_{s}$ is called the intersection multiplicity of $D_{1},\dots,D_{s}$ at $Z$ and it coincides with the multiplicity $\operatorname{mult}_{Z}(D_{1}\cdot\ldots\cdot D_{s})$ defined in Definition 2.6 (see [Ful98, Example 7.1.10]).

We have the following well-known lemma. Below, a general hyperplane on a germ of a regular closed point $P\in X$ is a Cartier divisor defined at $P$ by an equation $f\in\mathfrak{m}_{X,P}$ such that its image in $\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^{2}$ is general.

Lemma 2.8**.**

Let $P\in X$ be a germ of a regular closed point of an algebraic $\Bbbk$-scheme, let $Z\in Z_{j}(X)_{\mathbb{Q}}$ be an effective rational $j$-cycle on $X$ and let $H_{1},\dots,H_{i}$ be general hyperplanes through $P$, where $i\leq j$ are nonnegative integers. Then,

[TABLE]

Proof.

If suffices to prove the case where $P$ is in the support of $Z$. Let $H_{1},\dots,H_{i}$ be general hyperplanes through $P$ so that $\operatorname{Supp}(Z)\cap H_{1}\cap\cdots\cap H_{i}$ has pure dimension $j-i$ after possibly shrinking $P\in X$. We take general hyperplanes $H_{i+1},\dots,H_{j}$ so that $\operatorname{Supp}(Z)\cap H_{1}\cap\cdots\cap H_{j}$ consists only of the point $P$ after possibly shrinking $P\in X$. Since $H_{1},\ldots,H_{j}$ are general, their strict transforms have empty intersection with the strict transform of $\operatorname{Supp}(Z)$ under the blowup at $P$. By [Ful98, Example 12.4.8] (or Theorem 3.1 for the usual blowup), we have

[TABLE]

Applying this result for the effective $(j-i)$-cycle $Z\cdot H_{1}\cdot\ldots\cdot H_{i}$, we obtain

[TABLE]

and the lemma follows. ∎

In Theorem 3.7, we use local analytic coordinate changes. The lemma below guarantees that this does not alter intersection numbers. In the lemma, we use $\hat{\mathcal{O}}_{X,P}$ to denote the completion of the local ring $\mathcal{O}_{X,P}$ and we use $f^{\leq k}$ to denote the truncation of a power series $f\in\Bbbk[[x_{1},\ldots,x_{d}]]$ up to degree $k$.

Lemma 2.9**.**

Let $d$ be a positive integer. Let $P\in X$ be a $\Bbbk$-rational point of a smooth integral $d$-dimensional algebraic $\Bbbk$-scheme. Let $\Psi\colon\hat{\mathcal{O}}_{X,P}\to\Bbbk[[x_{1},\ldots,x_{d}]]$ be a $\Bbbk$-algebra isomorphism. Let $D_{1},\ldots,D_{d}$ be effective Cartier divisors on $X$ such that $P$ is an isolated point of $\operatorname{Supp}D_{1}\cap\ldots\cap\operatorname{Supp}D_{d}$. Let $f_{1},\ldots,f_{d}\in\Bbbk[[x_{1},\ldots,x_{d}]]$ be the images under $\Psi$ of chosen local equations of respectively $D_{1},\ldots,D_{d}$. Then, for every sufficiently large positive integer $N$, the origin $\bm{0}$ is an isolated point of the intersection of the supports of the divisors $V(f_{1}^{\leq N}),\ldots,V(f_{d}^{\leq N})$ on $\mathbb{A}_{\Bbbk}^{d}$ and

[TABLE]

Proof.

Let $F_{1},\ldots,F_{d}$ denote the chosen local equations of respectively $D_{1},\ldots,D_{d}$. Since $X$ is regular, by [Ful98, Example 7.1.10(a)], the intersection multiplicity is computed by the length of a $\mathcal{O}_{X,P}$-module as below,

[TABLE]

As written in [Ful98, Example 7.1.10(b)], modules of finite length are not altered by completion. We find

[TABLE]

By [Liu06, Exercise 7.1.6(d)], we have

[TABLE]

Let $N_{0}$ be a positive integer such that $(x_{1},\ldots,x_{d})^{N_{0}}\subseteq(f_{1},\ldots,f_{d})$. Then, for every positive integer $N\geq N_{0}$, the origin $\bm{0}$ is an isolated point of the intersection of the divisors $V(f_{1}^{\leq N}),\ldots,V(f_{d}^{\leq N})$ on $\mathbb{A}_{\Bbbk}^{d}$ and

[TABLE]

By [Ful98, Example 7.1.10(b)],

[TABLE]

Definition 2.10**.**

Let $X$ be an algebraic $\Bbbk$-scheme and let $P\in X$ be a closed point. For an effective Cartier divisor $D$ on $X$, we define

[TABLE]

where $f\in\mathcal{O}_{X,P}$ is the local equation of $D$. In general, for a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D=\sum a_{i}D_{i}$, where $a_{i}\in\mathbb{Q}$ and $D_{i}$ are effective Cartier divisors, we define

[TABLE]

and call it the order of $D$ at $P$.

Note that $\operatorname{ord}_{P}(D)=\operatorname{mult}_{P}(D)$ if $X$ is regular at $P$.

The following result is useful when applying Theorem 3.8.

Lemma 2.11**.**

Let $X$ be a normal $d$-dimensional integral scheme that is proper over an arbitrary field $\Bbbk$ with $d\geq 3$. Let $D_{1},D_{2},S_{1},\dots,S_{d-3}$ be effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors on $X$ that intersect properly, that is, any irreducible component of

[TABLE]

is a curve, and let $P$ be a $\Bbbk$-point of $X$ such that $P\in X$ is a local complete intersection singularity. If $T$ is an effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$ such that it is nef and $\operatorname{Supp}(T)$ does not contain any irreducible component of $\Gamma$ passing through $P$, then

[TABLE]

Proof.

Replacing $D_{1},D_{2},S_{1},\dots,S_{d-3}$ and $T$ by their positive multiples, we may assume that they are Cartier divisors. We decompose the effective $1$-cycle

[TABLE]

where $C$ and $C^{\prime}$ are effective $1$-cycles on $X$ such that any curve in $\operatorname{Supp}(C)$ passes through $P$ and no curve in $\operatorname{Supp}(C^{\prime})$ passes through $P$. We then obtain

[TABLE]

where the first inequality holds since $T$ is nef and the second inequality holds since $T\cap C$ is a finite set of points.

From here on, we work locally around $P$. In a neighborhood of $P$, $X$ can be identified with the complete intersection $\mathcal{X}_{1}\cap\cdots\cap\mathcal{X}_{c}$ in a smooth $(d+c)$-dimensional integral scheme $\mathcal{X}$ of finite type over $\Bbbk$, where $\mathcal{X}_{i}$ are hypersurfaces in $\mathcal{X}$. Note that $P$ is necessarily a smooth point of $\mathcal{X}$ since it is a regular $\Bbbk$-point of $\mathcal{X}$. We can take Cartier divisors $\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{S}_{1},\dots,\mathcal{S}_{d-3}$ and $\mathcal{T}$ on $\mathcal{X}$ such that $\mathcal{D}_{i}|_{X}=D_{i}$ for $i=1,2$, $\mathcal{S}_{j}|_{X}=S_{j}$ for $i=1,\dots,d-3$ and $\mathcal{T}|_{X}=T$. We may assume that $\operatorname{ord}_{P}(S_{j})=\operatorname{ord}_{P}(\mathcal{S}_{j})=\operatorname{mult}_{P}(\mathcal{S}_{j})$ for $j=1,\dots,d-3$ and $\operatorname{ord}_{P}(T)=\operatorname{ord}_{P}(\mathcal{T})=\operatorname{mult}_{P}(\mathcal{T})$. Then we have

[TABLE]

where the inequality follows from [Ful98, Corollary 12.4] since $P\in\mathcal{X}$ is a smooth point. The assertion follows from (2.2) and (2.3). ∎

The method of maximal singularities for Fano 3-folds

In this section, the ground field is assumed to be $\mathbb{C}$. We recall definitions concerning maximal singularities and explain the characterization of birational rigidity of Fano $3$-folds in terms of maximal singularities. Throughout the paper, a Fano variety is a normal projective $\mathbb{Q}$-factorial variety with only terminal singularities whose anticanonical divisor is ample. By a divisorial contraction $\varphi\colon Y\to X$, we mean a proper birational morphism between varieties with only terminal singularities such that $-K_{Y}$ is $\varphi$-ample and that the exceptional locus $E$ is a prime divisor on $Y$.

Let $X$ be a Fano $3$-fold of Picard number $1$.

Definition 2.12**.**

A divisorial contraction $\varphi\colon Y\to X$ is called a maximal extraction if there is a mobile linear system $\mathcal{M}\sim_{\mathbb{Q}}-nK_{X}$, where $n>0$ is a rational number, such that the pair $(X,\frac{1}{n}\mathcal{M})$ is not canonical along $E$, that is, the inequality $\operatorname{mult}_{E}(\mathcal{M})>na_{E}(X)$ holds, where $a_{E}(X):=\operatorname{ord}_{E}K_{Y/X}$ is the discrepancy of $X$ along $E$. The center of a maximal extraction is called a maximal center.

Definition 2.13**.**

A birational automorphism $\sigma\colon X\dashrightarrow X$ is called an elementary self-link or a Sarkisov self-link if there is a divisorial contraction $\varphi\colon Y\to X$, a birational map $\tau\colon Y\dashrightarrow Y^{\prime}$ which is a composite of inverse flips, a flop and flips between normal projective $\mathbb{Q}$-factorial $3$-folds with only terminal singularities and a divisorial contraction $\varphi^{\prime}\colon Y^{\prime}\to X$ such that the diagram

[TABLE]

commutes. In this case, we say that the elementary self-link $\sigma$ is initiated by $\varphi$.

The following is a characterization of birational rigidity which is well-known to experts. We refer the reader to [Oka18, Lemma 2.34] for the proof (of a more general result).

Theorem 2.14**.**

Let $X$ be a Fano $3$-fold of Picard number $1$. Then $X$ is birationally rigid if and only if, for each divisorial contraction $\varphi\colon Y\to X$,

• •

either $\varphi$ is not a maximal extraction,

• •

or there exists an elementary self-link initiated by $\varphi$.

Divisorial contractions to $cA_{k}$-points

In this section, the ground field is assumed to be $\mathbb{C}$. We recall the classification of divisorial contractions with center an isolated $cA_{k}$ point.

Definition 2.15**.**

Let $k$ be a positive integer. A $3$-fold germ $P\in X$ is called a $cA_{k}$-singularity (or a $cA_{k}$-point) if its general hyperplane section $P\in S$ is a Du Val singularity of type $A_{k}$.

It is known that a $cA_{k}$ singularity is terminal if and only if it is isolated.

Theorem 2.16** ([Kaw03, Theorem 1.13], [Kaw02, Theorem 1.1], [Yam18, Theorem 2.6] and [Pae24b, Theorem 6.10]).**

Let $P\in X$ be an isolated $cA_{k}$-point, where $k\geq 1$, and let $\varphi$ be a divisorial contraction with center $P\in X$. Then one of the following holds.

1. (1)

After locally analytically identifying $P\in X$ with the germ

[TABLE]

$\varphi$* is the weighted blow-up with weights $\operatorname{wt}(x_{1},x_{2},x_{3},x_{4})=(r_{1},r_{2},a,1)$, where $r_{1}+r_{2}=(k+1)a$ and $a$ is co-prime to $r_{1}$ and $r_{2}$, $g$ has weighted order $r_{1}+r_{2}$ with weights $\operatorname{wt}(x_{1},x_{4})=(a,1)$, and the monomial $x_{3}^{(r_{1}+r_{2})/a}\in g$.* 2. (2)

The singularity $P\in X$ is of type $cA_{1}$ and, after locally analytically identifying $P\in X$ with the germ

[TABLE]

$\varphi$* is the weighted blow-up with weights $\operatorname{wt}(x_{1},x_{2},x_{3},x_{4})=(1,5,3,2)$.* 3. (3)

The singularity $P\in X$ is of type $cA_{2}$ and, after locally analytically identifying $P\in X$ with the germ

[TABLE]

$\varphi$* is the weighted blow-up with weights $\operatorname{wt}(x_{1},x_{2},x_{3},x_{4})=(4,3,2,1)$.*

Remark 2.17*.*

Let $\varphi$ be a divisorial contraction with center a $cA_{k}$ point $P\in X$. If $\varphi$ is the one described in (1) (resp. (2), resp. (3)) of Theorem 2.16, then its discrepancy is $a$ (resp. $4$, resp. $3$).

We introduce a subclass of $cA_{k}$-singularity, called an ordinary $cA_{k}$-singularity that will be necessary in section 5.

Definition 2.18**.**

Let $P\in X$ be an isolated $cA_{k}$-point and $\varphi\colon Y\to X$ be a divisorial contraction with center $P$. The weights of $\varphi$ is defined to be $(r_{1},r_{2},a,1)$ (resp. $(1,5,3,2)$, resp. $(4,3,2,1)$) if $\varphi$ corresponds to (1) (resp. (2), resp. (3)) of Theorem 2.16. A divisorial contraction $\varphi$ is called exceptional if it corresponds to either (2) or (3) of Theorem 2.16.

Definition 2.19**.**

Let $P\in X$ be an isolated $cA_{k}$ point. We say that $P\in X$ is ordinary if it does not admit a divisorial contraction of discrepancy greater than $1$ that is not exceptional.

Divisorial contractions to an isolated ordinary $cA_{k}$ point is either a divisorial contraction described in (1) of Theorem 2.16 with $a=1$ or an exceptional divisorial contraction. Most of the isolated $cA_{2}$ points are ordinary as the following remarks suggest.

Remark 2.20*.*

Let $P\in X$ be an isolated $cA_{2}$ point defined in $\mathbb{C}^{4}$ by the equation

[TABLE]

where $f$ has order $3$. If the degree $3$ homogeneous part of $f$ is not the cube of a linear form in $z$ and $t$, then $P\in X$ is ordinary.

Remark 2.21*.*

Let $P\in X$ be the isolated $cA_{2}$ point defined in $\mathbb{A}^{4}$ by the equation

[TABLE]

where $m\geq 3$. Then it is ordinary if and only if $m\leq 5$. If $m=3$, then $P\in X$ does not admit a divisorial contraction of exceptional type, so that it admits 2 divisorial contractions of discrepancy $1$. If $m=4$, then $P\in X$ is equivalent to

[TABLE]

and, with this identification, the weighted blowup with weights $\operatorname{wt}(x,y,z,t)=(4,3,2,1)$ is a divisorial contraction of exceptional type.

Local inequalities

Main inequality

Theorem 3.1 is a generalization of [FL82, Theorem 3] and [Ful98, Example 12.4.8] to weighted blowups with respect to a regular system of parameters.

Theorem 3.1**.**

Let $P$ be a closed point of a $d$-dimensional regular integral algebraic $\Bbbk$-scheme $X$, where $\Bbbk$ is a field and $d\geq 2$. Let ${\bm{w}}=(w_{1},\ldots,w_{d})$ be a tuple of positive integers such that $\gcd{\bm{w}}=1$. Let $\varphi\colon Y\to X$ be the ${\bm{w}}$-blowup with respect to a regular system of parameters at $P$ and let $E$ be the exceptional prime divisor. Let $D_{1},\ldots,D_{d}$ be effective $\mathbb{Q}$-divisors on $X$ such that $P$ is an isolated point of $\operatorname{Supp}D_{1}\cap\ldots\cap\operatorname{Supp}D_{d}$. For every $i$, let $\tilde{D}_{i}$ be the strict transform of $D_{i}$. Then, both of the following hold:

[TABLE]

where $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ denotes the degree of the [math]-cycle class $\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d}$ with coefficients in $\mathbb{Q}$ on a small neighborhood of $E$ and $[\kappa(P):\Bbbk]$ is the degree of the field extension. Moreover, $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ is zero if and only if $\operatorname{Supp}\tilde{D}_{1}\cap\ldots\cap\operatorname{Supp}\tilde{D}_{d}\cap E$ is empty.

Proof.

We follow the proof of [FL82, Theorem 3]. By replacing $D_{i}$ with their positive multiples, we may assume that $D_{i}$ are Cartier divisors. Let $b$ be the least common multiple of $w_{1},\ldots,w_{d}$. Let $x_{1},\ldots,x_{d}$ be the regular system of parameters at $P$ along which the weighted blowup was taken. By [Ful98, Example 2.4.2], we can replace $X$ with a suitable open neighborhood of $P$ such that $D_{1}\cap\ldots\cap D_{d}=\{P\}$ and such that $x_{1},\ldots,x_{d}$ are defined on $X$. For every Weil divisor $D$ on $Y$, $bD$ is Cartier. Let $\eta\colon E\to P$ be the restriction of $\varphi$. Using the projection formula [Ful98, Proposition 2.3], both of the following hold:

1. (i)

$\eta_{*}(\varphi^{*}D_{1}\cdot\ldots\cdot\varphi^{*}D_{d})=D_{1}\cdot\ldots\cdot D_{d}$, 2. (ii)

for all $j\in\{1,\ldots,d-1\}$, $\eta_{*}(\varphi^{*}D_{1}\cdot\ldots\cdot\varphi^{*}D_{j}\cdot E^{d-j})=0$.

Denote $m_{i}:=v_{E}(D_{i})$. Equivalently,

1. (iii)

for all $i\in\{1,\ldots,d\}$, $\varphi^{*}(D_{i})=\tilde{D_{i}}+m_{i}E$.

Let $H_{i}$ be the Weil divisor $V(x_{i})$ on $X$ and let $\tilde{H}_{i}$ denote the strict transform of the divisor $H_{i}$ on $Y$. For all $i$, we have

[TABLE]

By (i)–(iii) and bilinearity of intersection products,

[TABLE]

Therefore,

[TABLE]

After again replacing $X$ with a suitable open neighborhood of $P$, we have $H_{1}\cdot\ldots\cdot H_{d}=[P]$ and we find

1. (iv)

\eta_{*}\bigl{(}(bE)^{d}\bigr{)}=\dfrac{(-1)^{d-1}b^{d}}{w_{1}\cdot\ldots\cdot w_{d}}[P].

By (i)–(iv) and bilinearity of intersection products,

[TABLE]

The intersection class $b\tilde{D}_{1}\cdot\ldots\cdot b\tilde{D}_{d}$ is a zero-cycle up to rational equivalence on $\tilde{D}_{1}\cap\ldots\cap\tilde{D}_{d}\subseteq E$. By [Ful98, Theorem 1.4 and Definition 1.4], all rationally equivalent zero-cycles $\alpha$ on $E$ have the same degree, which coincides with the degree of $\eta_{*}(\alpha)$. This proves equation 3.1.

Now we prove inequality 3.2. Let $L$ be the ample line bundle $\left.\mathcal{O}_{Y}(-bE)\right|_{E}$. We shrink $X$ if necessary so that the divisors $D_{1},\ldots,D_{d}$ are principal. By (iii), $\mathcal{O}(\tilde{D}_{i})=\mathcal{O}(-m_{i}E)$. Since every $b\tilde{D}_{i}$ is an effective Cartier divisor of $Y$, the closed embedding

[TABLE]

is a regular embedding of codimension $d$. By [Ful98, Appendix B.6.2 or B.7.1] or [Liu06, Corollary 6.3.8], the normal sheaf to $b\tilde{D}_{1}\times\ldots\times b\tilde{D}_{d}$ in $Y\times\ldots\times Y$ is a vector bundle $\mathcal{N}$ of rank $d$ on $b\tilde{D}_{1}\times\ldots\times b\tilde{D}_{d}$. Consider the commutative diagram

[TABLE]

where $Y\to Y\times\ldots\times Y$ is the diagonal embedding and $b\tilde{D}_{1}\cap\ldots\cap b\tilde{D}_{d}$ is the scheme-theoretic intersection. By [Ful98, Example 12.4.8], the intersection class $b\tilde{D}_{1}\cdot\ldots\cdot b\tilde{D}_{d}$ coincides with $s^{*}[C_{b\tilde{D}_{1}\cap\ldots\cap b\tilde{D}_{d}}Y]$, where

[TABLE]

is the zero-section and $C_{b\tilde{D}_{1}\cap\ldots\cap b\tilde{D}_{d}}Y$ is the normal cone to $b\tilde{D}_{1}\cap\ldots\cap b\tilde{D}_{d}$ in $Y$. We have

[TABLE]

Note that $L$ is a globally generated line bundle. Therefore, $L^{\otimes m_{i}}\otimes L^{\vee}$ is generated by global sections. Since a finite direct sum of globally generated sheaves is globally generated, we find that

[TABLE]

is globally generated. By [Ful98, Theorem 12.1(b)], we have

[TABLE]

where $W_{1},\ldots,W_{q}$ are the distinct irreducible (possibly nonreduced) components of $s^{-1}(C_{b\tilde{D}_{1}\cap\ldots\cap b\tilde{D}_{d}}Y)$, where $q$ is a nonnegative integer.

Lastly, note that if $\tilde{D}_{1}\cap\ldots\cap\tilde{D}_{d}$ is empty, then $(b\tilde{D}_{1})\cdot\ldots\cdot(b\tilde{D}_{d})$ is a cycle class on the empty scheme, therefore its degree is zero. Conversely, if $(b\tilde{D}_{1})\cdot\ldots\cdot(b\tilde{D}_{d})$ is zero, then $s^{-1}(C_{b\tilde{D}_{1}\cap\ldots\cap b\tilde{D}_{d}}Y)$ is empty, therefore $\tilde{D}_{1}\cap\ldots\cap\tilde{D}_{d}$ is empty. ∎

Remark 3.2*.*

If $\bm{w}=(1,\ldots,1)$ in Theorem 3.1, then we recover the classical intersection theory result. There is an error in the formula in [Ful98, Corollary in Example 12.4.8], namely the factor $1/{[\kappa(P):\Bbbk]}$ is missing in front of $\sum_{i=1}^{t}m_{i}\deg(\alpha_{i})$ and $\sum_{i=1}^{t}m_{i}\deg(Z_{i})$. The point $P$ is $\Bbbk$-rational in [Ful98, Corollary 12.4] and [FL82, Theorem 3], so that $[\kappa(P):\Bbbk]=1$.

Intersection cycles of mobile linear systems around LCI singularities

Let $c\geq 0$, $d\geq 2$ be integers and let $\Bbbk$ be a field. Let $P\in\mathcal{X}$ be the germ of a regular point on an algebraic $\Bbbk$-scheme of dimension $d+c$ with local coordinates $x_{1},\dots,x_{d+c}$ at $P$, and let

[TABLE]

be a local complete intersection singularity of dimension $d$, where $f_{1},\dots,f_{c}\in\mathcal{O}_{\mathcal{X},P}$ is a regular sequence. We assume that $X$ is normal, that is, it is regular in codimension $1$. We choose a tuple ${\bm{w}}:=(w_{1},\dots,w_{d+c})$ of positive integers such that $\gcd{\bm{w}}:=\gcd(w_{1},\dots,w_{d+c})=1$. We consider the ${\bm{w}}$-weights on the coordinates $x_{1},\dots,x_{d+c}$, that is,

[TABLE]

Let $\Phi\colon\mathcal{Y}\to\mathcal{X}$ be the weighted blow-up of $\mathcal{X}$ at $P\in\mathcal{X}$ with weights $\operatorname{wt}(x_{1},\dots,x_{d+c})={\bm{w}}$ and let $\mathcal{E}$ be its exceptional divisor. Let $Y$ be the strict transform of $X$ via $\Phi$ and let $\varphi$ be the restriction $\Phi|_{Y}$. Let $E$ be the exceptional divisor of $\varphi$. We consider the following condition.

Condition 3.3**.**

The closed subscheme

[TABLE]

is an integral scheme of dimension $d-1$, where $\mathbb{P}({\bm{w}})=\mathbb{P}(w_{1},\dots,w_{d+c})$ is the weighted projective space with homogeneous coordinates $x_{1},\dots,x_{d+c}$ of weights $w_{1},\dots,w_{d+c}$, respectively.

Lemma 3.4**.**

Under the above setting, we assume that Condition 3.3 is satisfied. Then the following assertions hold.

1. (1)

$\mathcal{E}|_{Y}=E$* and it is a prime divisor.* 2. (2)

The variety $Y$ is normal. 3. (3)

For any Cartier divisor $D$ on $X$, there is a Cartier divisor $\mathcal{D}$ on $\mathcal{X}$ such that $\mathcal{D}|_{X}=D$ and $v_{\mathcal{E}}(\mathcal{D})=v_{E}(D)$.

Proof.

The scheme-theoretic intersection $\mathcal{E}\cap Y$ is isomorphic to the closed subscheme defined in (3.3) and $E$ coincides with them set-theoretically. The scheme $\mathcal{E}\cap Y$ is integral by Condition 3.3. It follows that we have $\mathcal{E}|_{Y}=E$ as divisors on $Y$. This proves (1).

We prove (2). The variety $Y$ is covered by Zariski open subsets $U_{i}$ for $i=1,\dots,d+c$, where $U_{i}$ is the quotient of a local complete intersection variety which is regular in codimension $1$ by an action of the cyclic group $\bm{\mu}_{w_{i}}$. In particular, $Y$ is normal since the quotient of a normal variety is normal.

We prove (3). We may assume that $D$ is effective. Let $\phi\in\mathcal{O}_{X,P}$ be the local equation of $D$ and let $g\in\mathcal{O}_{\mathcal{X},P}$ be such that its residue class in $\mathcal{O}_{X,P}\cong\mathcal{O}_{\mathcal{X},P}/(f_{1},\dots,f_{c})$ is $\phi$. Possibly replacing $g$, we may assume that $g^{\bm{w}}\notin(f_{1}^{\bm{w}},\dots,f_{c}^{\bm{w}})$. We set $\mathcal{D}:=\mathrm{div}(g)$, which is a Cartier divisor on $\mathcal{X}$. We have

[TABLE]

The restriction $(\Phi_{*}^{-1}\mathcal{D})|_{\mathcal{E}}$ is isomorphic to the closed subscheme $(g^{\bm{w}}=0)$ in $\mathcal{E}$, where we identify $\mathcal{E}$ with the weighted projective space $\mathbb{P}({\bm{w}})$ with homogeneous coordinates $x_{1},\dots,x_{d+c}$ of weights $w_{1},\dots,w_{d+c}$, respectively. It follows that $\Phi_{*}^{-1}\mathcal{D}$ does not contain $E$, and hence $(\Phi_{*}^{-1}\mathcal{D})|_{Y}=\varphi_{*}^{-1}D$. By restricting the equality (3.4) to $Y$, we obtain

[TABLE]

since $\mathcal{E}|_{Y}=E$. This shows $v_{\mathcal{E}}(\mathcal{D})=v_{E}(D)$. ∎

Theorem 3.5**.**

Under the above setting, we assume that $w_{1}\leq w_{2}\leq\cdots\leq w_{d+c}$ and that Condition 3.3 is satisfied. Suppose that there is a mobile linear system $\mathcal{M}$ of Cartier divisors on $X$ and a rational number $n>0$ such that the pair $(X,\frac{1}{n}\mathcal{M})$ is not canonical at $E$. Then, for general members $D_{1},D_{2}\in\mathcal{M}$, we have

[TABLE]

Proof.

By Lemma 3.4, we can take divisors $\mathcal{D}_{i}$ on $\mathcal{X}$ such that $\mathcal{D}_{i}|_{X}=D_{i}$ and $v_{\mathcal{E}}(\mathcal{D}_{i})=v_{E}(D_{i})$ for $i=1,2$. As a $(d-2)$-cycle on $\mathcal{X}$, we have

[TABLE]

where $\mathcal{X}_{i}:=(f_{i}=0)\subset\mathcal{X}$. Let $\mathcal{H}_{1},\dots,\mathcal{H}_{d-2}$ be general hyperplanes on $\mathcal{X}$ passing through $P$ so that $P$ is an isolated component of

[TABLE]

We may assume that $v_{\mathcal{E}}(\mathcal{H}_{i})\geq w_{i}$ for $i=1,\dots,d-2$. By the assumption that $(X,\frac{1}{n}\mathcal{M})$ is not canonical at $E$, we obtain

[TABLE]

Thus, we have

[TABLE]

where the inequality follows from Theorem 3.1. ∎

We have a formula for computing the discrepancy $a_{E}(X):=\operatorname{ord}_{E}K_{Y/X}$ under some additional assumptions.

Lemma 3.6**.**

Under the above setting, we assume that Condition 3.3 is satisfied. Assume in addition that either $c=0$ or $c\geq 1$ and the weighted complete intersection (3.3) is well-formed (See §6.1 for the definition of well-formedness). Then we have

[TABLE]

Proof.

We set $\mathcal{X}_{i}:=(f_{i}=0)\subset\mathcal{X}$ and $X_{i}:=\mathcal{X}_{1}\cap\cdots\cap\mathcal{X}_{i}$ for $i=1,\dots,c$. We also set $\varphi_{i}=\Phi|_{Y_{i}}\colon Y_{i}\to X_{i}$, where $Y_{i}$ be the proper transform of $X_{i}$ on $\mathcal{Y}$. Let $E_{i}$ be the $\varphi_{i}$-exceptional divisor. We set $X_{0}:=\mathcal{X}$, $Y_{0}:=\mathcal{Y}$, $\varphi_{0}:=\Phi\colon Y_{0}\to X_{0}$ and $E_{0}:=\mathcal{E}$. We have the formula

[TABLE]

which in particular proves the assertion when $c=0$. We proceed the proof by assuming $c\geq 1$.

We choose and fix any $i\in\{0,1,\dots,c\}$. We see that the closed subscheme

[TABLE]

is integral because otherwise the closed subscheme (3.3) cannot be integral and this is impossible by Condition 3.3. We can apply Lemma 3.4 to the local complete intersection singularity $P\in X_{i}$ and we see that $E_{i}=\mathcal{E}|_{Y_{i}}$ is a prime divisor that coincides with the weighted complete intersection (3.6) and that $Y_{i}$ is a normal variety. The divisor $X_{i+1}$ is a Cartier divisor on $X_{i}$ for any $i$. The exceptional divisor $\mathcal{E}$ is a Cartier divisor on $\mathcal{Y}\setminus\operatorname{Sing}(\mathcal{E})$, and hence $E_{i}=\mathcal{E}\cap Y_{i}$ is a Cartier divisor on $Y_{i}\setminus\operatorname{Sing}(\mathcal{E})\cap E_{i}$. The closed set $\operatorname{Sing}(\mathcal{E})\cap E_{i}$ is of codimension at least $3$ in $Y_{i}$ since $E_{i}\subset\mathcal{E}\cong\mathbb{P}({\bm{w}})$ is well-formed. It follows that $Y_{i+1}=\varphi_{i}^{*}X_{i}-{\bm{w}}(f_{i})E_{i}$ is a Cartier divisor away from a closed subset of codimension at least $3$ in $Y_{i}$. By the adjunction formula, we have $(K_{X_{i}}+X_{i+1})|_{X_{i+1}}=K_{X_{i+1}}$ and $(K_{Y_{i}}+Y_{i+1})|_{Y_{i+1}}=K_{Y_{i+1}}$. By applying adjunction formulae successively for $X=X_{c}\subset\cdots\subset X_{0}$ and $Y=Y_{c}\subset\cdots\subset Y_{0}$ starting with (3.5), we obtain the desired formula. ∎

Local inequality for $cA_{k}$-singularities

In this section, the ground field is assumed to be $\mathbb{C}$.

Theorem 3.7**.**

Let $P\in X$ be an isolated $cA_{k}$ singularity, where $k\geq 1$, let $\mathcal{M}$ be a mobile linear system of Cartier divisors on $X$ and let $n>0$ be a rational number. Let $\varphi\colon Y\to X$ be a divisorial contraction with center $P$ and let $D_{1},D_{2}\in\mathcal{M}$ be general members. Suppose that the pair $(X,\frac{1}{n}\mathcal{M})$ is not canonical at the exceptional divisor $E$.

1. (1)

If $\varphi$ is of non-exceptional type and its weight is $(r_{1},r_{2},a,1)$, where $\gcd(a,r_{1})=\gcd(a,r_{2})=1$ and $r_{1}+r_{2}=(k+1)a$, then

[TABLE] 2. (2)

If $k=1$ and $\varphi$ is of exceptional type, then

[TABLE] 3. (3)

If $k=2$ and $\varphi$ is of exceptional type, then

[TABLE]

Proof.

We use Lemma 2.9 to move to a local analytic coordinate system. The weights of $\varphi$ is $(1,5,3,2)$ (resp. $(4,3,2,1)$) if we are in case (2) (resp. (3)). In any case, Condition 3.3 is satisfied and the assertion follows from Theorem 3.5. ∎

Theorem 3.8** (= Theorem 1.3).**

Let $P\in X$ be an isolated $cA_{k}$ singularity, where $k\geq 1$, let $\mathcal{M}$ be a mobile linear system of Cartier divisors and $n>0$ a rational number. Suppose that $P$ is a non-canonical center of the pair $(X,\frac{1}{n}\mathcal{M})$. Then for general members $D_{1},D_{2}\in\mathcal{M}$, we have

[TABLE]

Proof.

For any positive integers $r_{1}$ and $r_{2}$, we have

[TABLE]

If $k=1$ (resp. $2$), then $16/5n^{2}>2n^{2}$ (resp. $9/4n^{2}>4/3n^{2}$). Thus, the assertion follows from Theorem 3.7. ∎

Algebraic corollary

Corollary 3.9 is an algebraic corollary of Theorem 3.1, similar to [Ful98, Example 12.4.9].

Corollary 3.9**.**

Let $d$ be a positive integer. Let $\Bbbk$ be an arbitrary field and let ${\bm{w}}=(w_{1},\ldots,w_{d})$ be a $d$-tuple of positive integers such that $\gcd{\bm{w}}=1$. We consider the weights $\operatorname{wt}(x_{1},\dots,x_{d})={\bm{w}}$ and let $f_{1},\ldots,f_{d}\in\Bbbk[[x_{1},\ldots,x_{d}]]$. Then

[TABLE]

Moreover, if $f_{1},\ldots,f_{d}$ are polynomials and

[TABLE]

is finite, then

[TABLE]

where $\deg_{\bm{w}}(f)$ denotes the ${\bm{w}}$-degree of a nonzero polynomial $f\in\Bbbk[x_{1},\ldots,x_{d}]$, meaning the greatest integer $k$ such that the quasihomogeneous weighted-degree $k$ part of $f$ is nonzero.

Proof.

First, we prove the lower bound. If the left-hand side is infinite, then the inequality holds trivially. Otherwise, there exists a positive integer $N$ such that $\mathfrak{m}^{N}\subseteq(f_{1},\ldots,f_{d})$, where $\mathfrak{m}$ is the maximal ideal $(x_{1},\ldots,x_{d})$. Therefore $\mathfrak{m}^{N}\subseteq(g_{1},\ldots,g_{d})$, where the $g_{i}$ are the truncations $(f_{i})_{\deg\leq N}\in\Bbbk[x_{1},\ldots,x_{k}]$ up to degree $N$. We find ${\bm{w}}(g_{i})\geq{\bm{w}}(f_{i})$ and

[TABLE]

Let $P$ be the origin in $X:=\mathbb{A}_{\Bbbk}^{d}$ and let $D_{i}:=V(g_{i})$. By [Ful98, Example 7.1.10(b)],

[TABLE]

The first inequality in the statement of Corollary 3.9 now follows from Theorem 3.1.

Next, we prove the upper bound. Assign weights $1,w_{1},\ldots,w_{d}$ respectively to the variables $x_{0},\ldots,x_{d}$. Let $F_{1},\ldots,F_{d}\in\Bbbk[x_{0},\ldots,x_{d}]$ be the quasihomogeneous polynomials such that for every $i\in\{1,\ldots,d\}$, $x_{0}$ does not divide $F_{i}$ and $F_{i}(1,x_{1},\ldots,x_{d})=f_{i}$. Let $V(F_{i})$ denote the subscheme in the weighted projective space $\mathbb{P}(1,w_{1},\ldots,w_{d})$ defined by $F_{i}$. Let $P$ be the point $[1,0,\ldots,0]$ in $\mathbb{P}(1,w_{1},\ldots,w_{d})$. Note that either $P$ is not in $V(F_{1},\ldots,F_{d})$ or $P$ is an isolated point of $V(F_{1},\ldots,F_{d})$. By [Ful98, Example 7.1.10(b)],

[TABLE]

Since each $V(F_{i})$ is $\mathbb{Q}$-linearly equivalent to $\frac{\operatorname{deg}_{\bm{w}}(f_{i})}{w_{i}}\cdot V(x_{i})$, we find

[TABLE]

It is left to prove that on the open subset $U:=\mathbb{P}(1,w_{1},\ldots,w_{d})\setminus\{P\}$, the degree of the intersection class $(V(F_{1})\cap U)\cdot\ldots\cdot(V(F_{d})\cap U)$ is nonnegative. As in the proof of inequality 3.2 in Theorem 3.1, this follows from the arguments in the proof of [FL82, Theorem 3] by applying [FL82, Theorem 2(B)]. ∎

Generalizations to (LCI-)quotient singularities

Throughout this section, we work over the field $\mathbb{C}$ of complex numbers.

Main inequality for cyclic quotient singularities

We consider a generalization of Theorem 3.1 to cyclic quotient singularities. Suppose that $d\geq 2$. Let $r>0$ and $a_{1},\dots,a_{d}\geq 0$ be integers.

We say that an action of $\bm{\mu}_{r}$ on a germ $P\in V$ of a smooth $d$-dimensional variety is of type $\frac{1}{r}(a_{1},\dots,a_{d})$ if there exist a regular system of parameters $x_{1},\dots,x_{d}$ of $V$ at $P$ such that the $\mathbb{C}$-algebra automorphism of $\mathcal{O}_{V,P}$ induced by $\zeta\in\bm{\mu}_{r}$ sends $x_{i}$ to $\zeta^{a_{i}}x_{i}$ for $i=1,\dots,d$ and for any $r$th root of unity $\zeta\in\bm{\mu}_{r}$, so that the $\mathbb{C}$-algebra isomorphism $\mathbb{C}[[x_{1},\dots,x_{d}]]\to\hat{\mathcal{O}}_{V,P}$ induced by the inclusion $\mathbb{C}[x_{1},\dots,x_{n}]\to\mathcal{O}_{V,P}$ is $\bm{\mu}_{r}$-equivariant, where the $\bm{\mu}_{r}$-action on $\mathbb{C}[[x_{1},\dots,x_{n}]]$ is defined by $x_{i}\mapsto\zeta^{a_{i}}x_{i}$ for $i=1,\dots,d$ and for any $r$th root of unity $\zeta\in\mathbb{C}$.

Let $P\in X$ be a germ of a cyclic quotient singular point of type $\frac{1}{r}(a_{1},\dots,a_{d})$, that is, the germ $P\in X$ is isomorphic to the germ

[TABLE]

where $\hat{P}\in\hat{X}$ is a germ of a smooth $d$-dimensional variety admitting a $\bm{\mu}_{r}$-action of type $\frac{1}{r}(a_{1},\dots,a_{d})$ and $P^{\prime}$ is the image of $\hat{P}$ by the quotient morphism $\hat{X}\to\hat{X}/\bm{\mu}_{r}$. We identify $P\in X$ with $P^{\prime}\in\hat{X}/\bm{\mu}_{r}$ and denote by $q_{X}\colon\hat{X}\to X$ the quotient morphism.

We assume that

[TABLE]

for every $j\in\{1,\ldots,d\}$, that is, $q_{X}\colon\hat{X}\to X$ is étale in codimension $1$. By convention, we allow $r=1$, corresponding to a smooth point $P\in X$.

Let ${\bm{w}}=(w_{1},\dots,w_{d})$ be a tuple of positive integers such that $w_{i}\equiv a_{i}\pmod{r}$ for $i=1,\dots,d$ and let $x_{1},\dots,x_{d}$ be a regular system of parameters of $\hat{X}$ at $\hat{P}$ with respect to which the action of $\bm{\mu}_{r}$ on $\hat{X}$ is linear of type $\frac{1}{r}(a_{1},\dots,a_{d})$. We give a description of weighted blowup $\varphi\colon Y\to X$ of $P\in X$ with weights $w_{1}/r,\dots,w_{d}/r$. Let $\hat{\varphi}\colon\hat{Y}\to\hat{X}$ be the ${\bm{w}}$-blowup of $\hat{P}\in\hat{X}$ with respect to the regular system of parameters $x_{1},\dots,x_{d}$ and let $\hat{E}\cong\mathbb{P}(w_{1},\dots,w_{d})$ be its exceptional divisor. The $\bm{\mu}_{r}$-action on $\hat{X}$ lifts to an action on $\hat{Y}$ and we denote by $Y$ its quotient $\hat{Y}/\bm{\mu}_{r}$. Then the restriction $\varphi:=\hat{\varphi}|_{Y}\colon Y\to X$ is called a weighted blowup of $P\in X$ with weights $w_{1}/r,\dots,w_{d}/r$ with respect to $x_{1},\dots,x_{d}$. Let $q_{Y}\colon\hat{Y}\to Y$ be the quotient morphism, which is a finite morphism of degree $r$ ramified along the $\varphi$-exceptional divisor $E$, and we have $q_{Y}^{*}E=r\hat{E}$. We obtain the diagram:

[TABLE]

Corollary 4.1**.**

Let the notation and assumptions be as above. Let $D_{1},\ldots,D_{d}$ be effective $\mathbb{Q}$-divisors on $X$ such that $P$ is an isolated point of $D_{1}\cap\ldots\cap D_{d}$. For every $i$, let $\tilde{D}_{i}$ be the strict transform of $D_{i}$ on $Y$. Then, both of the following hold:

[TABLE]

where $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ denotes the degree of the [math]-cycle class $\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d}$ on a small neighborhood of $E$. Moreover, $\deg_{E}(\tilde{D}_{1}\cdot\ldots\cdot\tilde{D}_{d})$ is zero if and only if $\tilde{D}_{1}\cap\ldots\cap\tilde{D}_{d}\cap E$ is empty.

Proof.

For every $i$, let $\hat{D}_{i}$ be the pullback of $D_{i}$ under $q_{X}$. By Theorem 3.1,

[TABLE]

with equality if and only if $\hat{\varphi}_{*}^{-1}\hat{D}_{1}\cap\ldots\cap\hat{\varphi}_{*}^{-1}\hat{D}_{n}\cap\hat{E}$ is empty. By [Ful98, section 1.4],

[TABLE]

We have $q_{Y}^{*}E=r\hat{E}$ as explained above. This implies

[TABLE]

The corollary follows. ∎

Intersection cycles of mobile linear systems around LCI-quotient singularities

We consider a generalization of Theorem 3.5 to cyclic quotients of local complete intersection singularities.

Let $c\geq 0$, $d\geq 2$, $r>0$ and $a_{1},\dots,a_{d+c}\geq 0$ be integers. Let $\hat{P}\in\hat{\mathcal{X}}$ be the germ of a smooth $(d+c)$-dimensional variety with local coordinates $x_{1},\dots,x_{d+c}$ at $\hat{P}$ and let

[TABLE]

be a local complete intersection singularity of dimension $d$, where $f_{1},\dots,f_{c}\in\mathcal{O}_{\hat{\mathcal{X}},\hat{P}}$ is a regular sequence. We consider the action of $\bm{\mu}_{r}$ of type $\frac{1}{r}(a_{1},\dots,a_{d+c})$ on $\hat{\mathcal{X}}$ with respect to the local coordinates $x_{1},\dots,x_{d+c}$. We assume that $f_{i}$ is semi-invariant with respect to the $\bm{\mu}_{r}$-action for $i=1,\dots,c$, so that $\bm{\mu}_{r}$ acts on $\hat{X}$. Let $X$ and $\mathcal{X}$ be the quotients of $\hat{X}$ and $\hat{\mathcal{X}}$ by the $\bm{\mu}_{r}$-action. We assume that $\hat{X}$ is normal, which implies that $X$ is also normal. We denote by $q_{X}\colon\hat{X}\to X$ and $q_{\mathcal{X}}\colon\hat{\mathcal{X}}\to\mathcal{X}$ be the quotient morphisms. We set $P:=q_{X}(\hat{P})=q_{\mathcal{X}}(\hat{P})\in X\subset\mathcal{X}$. We assume that

[TABLE]

for every $j\in\{1,\ldots,d+c\}$. If $c=0$, then $\hat{X}=\hat{\mathcal{X}}$ and the singularity $P\in X$ is simply a quotient singularity of type $\frac{1}{r}(a_{1},\dots,a_{d})$. We allow $r=1$, corresponding to the case of a local complete intersection singularity.

Let $w_{1},\dots,w_{d+c}$ be positive integers such that $\gcd(w_{1},\dots,w_{d+c})=1$ and $w_{i}\equiv a_{i}\pmod{r}$ for $i=1,\dots,d+c$. We set ${\bm{w}}:=(w_{1},\dots,w_{d+c})$ and ${\bm{w}}_{r}:=\frac{1}{r}{\bm{w}}=\frac{1}{r}(w_{1},\dots,w_{d+c})$. Let $\varphi\colon Y\to X$ be a weighted blow-up of $P\in X$ with weights $\operatorname{wt}(x_{1},\dots,x_{d+c})={\bm{w}}_{r}$. Let $\hat{\Phi}\colon\hat{\mathcal{Y}}\to\hat{\mathcal{X}}$ be the weighted blow-up of $\hat{\mathcal{X}}$ at the origin $\hat{P}$ with weights $\operatorname{wt}(x_{1},\dots,x_{d+c})={\bm{w}}$. The $\bm{\mu}_{r}$-action on $\hat{\mathcal{X}}$ lifts to $\hat{\mathcal{Y}}$, and we obtain the induced morphism $\Phi\colon\mathcal{Y}\to\mathcal{X}$, where $\mathcal{Y}$ and $\mathcal{X}$ are the quotients of $\hat{\mathcal{Y}}$ and $\hat{\mathcal{X}}$, respectively, by the $\bm{\mu}_{r}$-action. We have the quotient morphism $q_{\mathcal{Y}}\colon\hat{\mathcal{Y}}\to\mathcal{Y}$, and the commutative diagram:

[TABLE]

We can identify $\hat{Y}$ and $Y$ with the proper transforms $\hat{\Phi}_{*}^{-1}\hat{X}\subset\hat{\mathcal{Y}}$ and $\Phi_{*}^{-1}X\subset\mathcal{Y}$, respectively. Under the above identifications, $\hat{\varphi}$ and $\varphi$ coincide with the restrictions $\hat{\Phi}|_{\hat{Y}}$ and $\Phi|_{Y}$. The diagram (4.1) restricts to the diagram:

[TABLE]

where $q_{Y}:=q_{\mathcal{Y}}|_{\hat{Y}}\colon\hat{Y}\to Y$ is the quotient morphism under the induced $\bm{\mu}_{r}$-action. Let $\hat{\mathcal{E}}$ and $\mathcal{E}$ be the exceptional divisors of $\hat{\Phi}$ and $\Phi$, respectively, both of which are isomorphic to $\mathbb{P}({\bm{w}})$. Note here that, by a slight abuse of notation, $\mathbb{P}({\bm{w}})=\mathbb{P}(w_{1},\dots,w_{d+c})$ is thought of as the weighted projective space with homogeneous coordinates $x_{1},\dots,x_{d+c}$ of weights $w_{1},\dots,w_{d+c}$, respectively. The quotient morphism $q_{\mathcal{Y}}\colon\hat{\mathcal{Y}}\to\mathcal{Y}$ is a finite morphism ramified along $\mathcal{E}$ and we have $q_{\mathcal{Y}}^{*}\mathcal{E}=r\hat{\mathcal{E}}$. We denote by $\hat{E}$ and $E$ the exceptional divisors of $\hat{\varphi}$ and $\varphi$, respectively. For an element $f\in\mathcal{O}_{\mathcal{X},P}$, we denote by ${\bm{w}}(f)$ (resp. ${\bm{w}}_{r}(f)$) the weighted order of $f$ with respect to the ${\bm{w}}$-weights (resp. ${\bm{w}}_{r}$-weights) and by $f^{\bm{w}}$ the least weight part of $f$ with respect to the ${\bm{w}}$-weights in the sense of Definition 2.2.

Lemma 4.2**.**

Under the above setting, we assume that Condition 3.3 is satisfied. Then the following assertions hold.

1. (1)

$\hat{\mathcal{E}}|_{\hat{Y}}=\hat{E}$* and $\mathcal{E}|_{Y}=E$, and they are both prime divisors.* 2. (2)

$Y$* is a normal variety.* 3. (3)

$\hat{E}$* is isomorphic to $E$ and we have $q_{Y}^{*}E=r\hat{E}$.* 4. (4)

If in addition either $c=0$ or $c\geq 1$ and the weighted complete intersection (3.3) is well-formed, then

[TABLE]

Proof.

By Lemma 3.4 applied to the local complete intersection singularity $\hat{P}\in\hat{X}$, we see that $\hat{\mathcal{E}}|_{\hat{Y}}=\hat{E}$ is a prime divisor and $\hat{Y}$ is normal. These imply that $\mathcal{E}|_{Y}=E$ and $Y$ is normal. Taking restriction of $q_{\mathcal{Y}}^{*}\mathcal{E}=r\hat{\mathcal{E}}$ to $Y$, we obtain $q_{Y}^{*}E=r\hat{E}$.

By Lemma 3.6 applied to $\hat{P}\in\hat{X}$, we have

[TABLE]

We see that $K_{\hat{X}}=q_{X}^{*}K_{X}$ and

[TABLE]

Combining these equalities, we obtain (4). ∎

Theorem 4.3**.**

Under the above setting, we assume that $w_{1}\leq w_{2}\leq\cdots\leq w_{d+c}$ and Condition 3.3 is satisfied. Suppose that there is a mobile linear system $\mathcal{M}$ on $X$ and a rational number $n>0$ such that $q_{X}^{*}\mathcal{M}$ is a linear system of Cartier divisors on $\hat{X}$ and the pair $(X,\frac{1}{n}\mathcal{M})$ is not canonical at the $\varphi$-exceptional divisor $E$. Then, for general members $D_{1},D_{2}$ of $\mathcal{M}$, we have

[TABLE]

Proof.

Let $D_{1},D_{2}\in\mathcal{M}$ be general members. By assumption, $\hat{D}_{i}:=q_{X}^{*}D_{i}$ is a Cartier divisor on $\hat{X}$, and hence we can take divisors $\hat{\mathcal{D}}_{i}$ on $\hat{\mathcal{X}}$ such that $\hat{\mathcal{D}}_{i}|_{X}=\hat{D}_{i}$ for $i=1,2$. By Lemma 3.4, we can choose $\hat{\mathcal{D}}_{i}$ so that $v_{\hat{E}}(\hat{D}_{i})=v_{\hat{\mathcal{E}}}(\hat{\mathcal{D}}_{i})$. As a $(d-2)$-cycle on the germ $\hat{\mathcal{X}}$, we have

[TABLE]

where $\hat{\mathcal{X}}_{i}:=(f_{i}=0)\subset\hat{\mathcal{X}}$ is a hypersurface for $i=1,\dots,c$. Let $\hat{\mathcal{H}}_{1},\dots,\hat{\mathcal{H}}_{d-2}$ be general hyperplanes of $\hat{\mathcal{X}}$ passing through $\hat{P}$ such that $\hat{P}$ is an isolated component of

[TABLE]

We may assume that $v_{\hat{\mathcal{E}}}(\hat{\mathcal{H}}_{i})\geq w_{i}$ for $i=1,2,\dots,d-2$. Moreover, we have

[TABLE]

where the last inequality follows from the assumption that $(X,\frac{1}{n}\mathcal{M})$ is not canonical at $E$. By Theorem 3.1, we have

[TABLE]

The desired inequality follows since $\operatorname{mult}_{P}(D_{1}\cdot D_{2})=\frac{1}{r}\operatorname{mult}_{\hat{P}}(\hat{D}_{1}\cdot\hat{D}_{2})$. ∎

Sextic double solids

The aim of this section is to prove the birational rigidity of factorial sextic double solids with at most $cA_{1}$-points and ordinary $cA_{2}$-points.

Birational involutions

This section is entirely devoted to the proof of the following result.

Proposition 5.1**.**

Let $X$ be a factorial sextic double solid admitting an isolated $cA_{2}$ point $P\in X$ and let $\varphi\colon Y\to X$ be a divisorial contraction of discrepancy $1$ with center $P$. Then, either $\varphi$ is not a maximal extraction or there is an elementary self-link $\sigma\colon X\dashrightarrow X$ initiated by $\varphi$.

Note that the singularity $P\in X$ is not necessarily ordinary in Proposition 5.1. Let $X$ be a factorial sextic double solid admitting a singular point $P\in X$ of type $cA_{2}$. Then, by [Pae24a, Theorem A], we may assume that $X\subset\mathbb{P}(1_{x},1_{y},1_{z},1_{t},3_{w})$ is defined by an equation of the form

[TABLE]

where $g_{i}=g_{i}(y,z,t)$ is a homogeneous polynomial of degree $i$ such that $g_{3}(y,z,0)\neq 0$, and $P=(1\!:\!0\!:\!0\!:\!0\!:\!0)$. We note that the expression $\mathbb{P}(1_{x},1_{y},1_{z},1_{t},3_{w})$ means that it is the weighted projective space $\mathbb{P}(1,1,1,1,3)$ with homogeneous coordinates $x,y,z,t,w$ of weights $1,1,1,1,3$, respectively. We introduce new homogeneous coordinates $\xi$ and $\zeta$ of weights $3$, and consider the relations:

[TABLE]

Eliminating $w=(\xi-\zeta)/2$, we identify $X$ with the complete intersection of codimension $2$ in $\mathbb{P}:=\mathbb{P}(1_{x},1_{y},1_{z},1_{t},3_{\xi},3_{\zeta})$ defined by the equations

[TABLE]

Let $\Phi\colon\mathcal{Y}\to\mathbb{P}$ be the weighted blow-up of $\mathbb{P}$ at $P=(1\!:\!0\!:\!\cdots\!:\!0)$ with weight $\operatorname{wt}(y,z,t,\xi,\zeta)=(1,1,1,1,2)$. We set $Y:=\Phi_{*}^{-1}X$ and $\varphi:=\Phi|_{Y}\colon Y\to X$. Let $\mathcal{E}\cong\mathbb{P}(1,1,1,1,2)$ be the exceptional divisor of $\Phi$ and let $E:=\mathcal{E}|_{Y}$ be the $\varphi$-exceptional divisor. Then we have isomorphisms

[TABLE]

We have $K_{Y}=\varphi^{*}K_{X}+E$ and $\varphi$ is a divisorial contraction of discrepancy $1$.

Note that $E^{3}=3/2$. We set $Q:=(0\!:\!\cdots\!:\!0\!:\!1)\in E$, which is a $\frac{1}{2}(1,1,1)$ point of $Y$. Let $\pi\colon X\dashrightarrow\mathbb{P}^{2}=\mathbb{P}(1_{y},1_{z},1_{t})$ be the projection to the coordinates $y,z,t$, and set $\rho:=\pi\circ\varphi\colon Y\dashrightarrow\mathbb{P}^{2}$.

Lemma 5.2**.**

The following assertions hold.

1. (1)

The rational map $\rho$ is defined outside the point $Q$. 2. (2)

The divisor $-K_{Y}$ is nef.

Proof.

We have

[TABLE]

which shows that $\pi$ is defined outside $P$. Hence the indeterminacy locus of $\rho$ is contained in $E$ and, under the isomorphism (5.2), it coincides with the set

[TABLE]

This proves (1).

The sections $y,z,t$ lift to sections of $-K_{Y}\sim-\varphi^{*}K_{X}-E$, and hence $\left|-K_{Y}\right|$ is base point free away from $Q$. This proves (2). ∎

Let $\psi\colon\hat{Y}\to Y$ be the Kawamata blow-up of $Y$ at $Q$. Then $\psi$ resolves the indeterminacy of $\rho$ and $\hat{\rho}:=\rho\circ\psi\colon\hat{Y}\to\mathbb{P}^{2}$ is everywhere defined. Note that $\hat{Y}$ is a smooth projective variety. The sections $y,z,t$ lift to sections of $-K_{\hat{Y}}$, and hence $\hat{\rho}$ is defined by the complete linear system $\left|-K_{\hat{Y}}\right|$. It is clear that general fibers of $\pi$ are irreducible and reduced. It follows that $\hat{\rho}$ has connected fibers, and hence it is an elliptic fibration. Let $F\cong\mathbb{P}^{2}$ be the $\psi$-exceptional divisor and set $\hat{E}:=\psi_{*}^{-1}E$. We have $\psi^{*}E=\hat{E}+\frac{1}{2}F$.

Lemma 5.3**.**

The following assertions hold.

1. (1)

The divisor $F$ is a section of $\hat{\rho}$ and $\hat{E}$ is a rational section of $\hat{\rho}$. 2. (2)

There are at most $3$ irreducible curves on $\hat{E}$ which are contracted by $\hat{\rho}$. 3. (3)

There is no prime divisor on $\hat{Y}$ that is mapped to a point by $\hat{\rho}$.

Proof.

A $\hat{\rho}$-fiber is rationally equivalent to $(-K_{Y})^{2}$ and we have

[TABLE]

Hence $\hat{E}$ is a rational $\hat{\rho}$-section and $F\cong\mathbb{P}^{2}$ is a $\hat{\rho}$-section.

Under the identification (5.2) of $E$, we further eliminate the variable $\xi=2t$, we have an isomorphism

[TABLE]

Then the map $\rho|_{E}\colon E\dashrightarrow\mathbb{P}^{2}$ is the projection to the coordinates $y,z,t$, and the morphism $\psi|_{\hat{E}}\colon\hat{E}\to E$ is the weighted blow-up of $E$ at the point $Q=(0\!:\!\cdots\!:\!0\!:\!1)$ with weight $\operatorname{wt}(y,z,t)=\frac{1}{2}(1,1,1)$. The irreducible curves on $\hat{E}$ that are contracted by $\hat{\rho}$ are exactly the proper transforms of the curves in $(t=g_{3}=0)\subset\mathbb{P}(1_{y},1_{z},1_{t},2_{\zeta})$. There are at most $3$ irreducible curves in the above set since $g(y,z,0)\neq 0$. This proves (2).

Suppose that there is a prime divisor $G$ on $\hat{Y}$ that is mapped to a point by $\hat{\rho}$. Then, $G\neq\hat{E},F$ by (1), and we see that $D:=\varphi_{*}\psi_{*}G$ is a prime divisor on $X$ that is mapped to a point by $\pi$. This is clearly impossible since a fiber of $\pi$ is a complete intersection of two distinct divisors in $|\mathfrak{m}_{P}(A)|$ and cannot be a divisor on $X$. This proves (3). ∎

Let $\hat{\chi}\colon\hat{Y}\dashrightarrow\hat{Y}$ be the reflection of the elliptic fibration with respect to the section $F$. The birational involution $\hat{\chi}$ is an isomorphism in codimension $1$ since $K_{Y}$ is $\hat{\rho}$-nef, and it descends to a birational involution $\chi\colon Y\dashrightarrow Y$ which is an isomorphism in codimension $1$ since $\hat{\chi}_{*}F=F$.

[TABLE]

Lemma 5.4**.**

If there are infinitely many reducible $\hat{\rho}$-fibers, then $\varphi$ is not a maximal extraction.

Proof.

Let $\Gamma_{\lambda}$, $\lambda\in\Lambda$, be reducible $\hat{\rho}$-fibers. Possibly removing finitely many elements from $\Lambda$, we may assume that, for any $\lambda\in\Lambda$, $\Gamma_{\lambda}$ does not contain any irreducible curve on $\hat{E}$ contracted by $\hat{\rho}$ (cf. (2) of Lemma 5.3).

We think of $\Gamma_{\lambda}$ as the $1$-cycle $\hat{\rho}^{*}\ell_{1}\cdot\hat{\rho}^{*}\ell_{2}$, where $\ell_{1}$ and $\ell_{2}$ are lines on $\mathbb{P}^{2}$ such that $\ell_{1}\cap\ell_{2}=\{\hat{\rho}(\Gamma)\}$. Note that $(-K_{Y}\cdot\Gamma_{\lambda})=0$ and $\Gamma_{\lambda}$ is an effective $1$-cycle with integral coefficients. Then we can write $\Gamma_{\lambda}=\Delta_{\lambda}+\Xi_{\lambda}$, where $\Delta_{\lambda}$ is an irreducible curve such that $(F\cdot\Delta_{\lambda})=1$ and $\Xi$ is an effective $1$-cycle with integral coefficients such that $\operatorname{Supp}(\Xi_{\lambda})\cap F=\emptyset$. We have $(-K_{\hat{Y}}\cdot(\Delta_{\lambda}+\Xi_{\lambda}))=0$ and $-K_{\hat{Y}}=-\psi^{*}K_{Y}-\frac{1}{2}E$. It follows that

[TABLE]

The effective $1$-cycle $\psi_{*}\Xi_{\lambda}$ is integral and its support does not pass through the unique non-Gorenstein singular point $Q$ of $Y$, and hence $(-K_{Y}\cdot\psi_{*}\Xi_{\lambda})\in\mathbb{Z}$. It follows that $(-K_{Y}\cdot\psi_{*}\Delta)=1/2$ and $(-K_{Y}\cdot\psi_{*}\Xi)=0$ since $-K_{Y}$ is nef by Lemma 5.2. By our choice of $\Gamma_{\lambda}$, no component of $\psi_{*}\Xi_{\lambda}\neq 0$ is contained in $E$. Hence $(E\cdot\psi_{*}\Xi)>0$ since $\varphi_{*}\psi_{*}\Xi$ is a nonzero effective $1$-cycle on $X$ and we have

[TABLE]

The intersection number of $-K_{Y}$ and any irreducible component of $\psi_{*}\Xi$ is [math] since $(-K_{Y}\cdot\psi_{*}\Xi_{\lambda})=0$ and $\psi_{*}\Xi_{\lambda}$ is effective. Thus, there is an irreducible component of $\psi_{*}\Xi_{\lambda}$, denoted by $\Theta_{\lambda}$, such that $(-K_{Y}\cdot\Theta_{\lambda})=0$ and $(E\cdot\Theta_{\lambda})>0$.

It follows that there are infinitely many irreducible curves $\{\Theta_{\lambda}\}$ on $Y$ that intersect $K_{Y}$ trivially and $E$ positively. By [Oka18, Lemma 2.20], $\varphi$ is not a maximal extraction. ∎

Lemma 5.5**.**

If there are at most finitely many reducible $\hat{\rho}$-fibers, then $\chi$ is not an isomorphism. In particular, the induced birational map $\sigma:=\varphi^{-1}\chi\circ\varphi\colon X\dashrightarrow X$ is an elementary self-link.

Proof.

Let $G$ be a prime divisor on $\hat{Y}$ which is $\hat{\rho}$-vertical, that is, $\hat{\rho}(G)$ is a proper closed subvariety of $\mathbb{P}^{2}$. By Lemma 5.3, $C:=\hat{\rho}(G)$ is an irreducible curve. If $\hat{\rho}^{*}C$ is reducible, then it contains a prime divisor $G^{\prime}\neq G$ in its support and we also have $\hat{\rho}(G^{\prime})=C$. This implies that the $\hat{\rho}$-fibers over any point of $C$ are reducible. This is impossible by assumption. It follows that any prime vertical divisor on $\hat{Y}$ is the pullback of an irreducible curve on $\mathbb{P}^{2}$. In particular, the sum of vertical divisors on $\hat{Y}$ is linearly equivalent to a multiple of $-K_{\hat{Y}}$.

Now suppose that $\chi$ is an isomorphism. Then $\chi_{*}E=E$ and, on the generic fiber $\hat{Y}_{\eta}$ of $\hat{\rho}$, we see that $(\hat{E}-F)|_{\hat{Y}_{\eta}}\in\operatorname{Pic}^{0}(\hat{Y}_{\eta})$ is a $2$-torsion. It follows that $\hat{E}-F$ is $\mathbb{Q}$-linearly equivalent to a sum of $\hat{\rho}$-vertical divisors. By what we have proved, we have $\hat{E}-F\sim_{\mathbb{Q}}mK_{\hat{Y}}$ for some $m\in\mathbb{Q}$. This is impossible since $\hat{E},F$ and $K_{\hat{Y}}$ generate $\operatorname{Pic}(\hat{Y})_{\mathbb{Q}}\cong\mathbb{Q}^{3}$. Therefore, $\chi$ is not an isomorphism. ∎

Proof of Proposition 5.1.

Let $X$ be a factorial sextic double solid and let $P\in X$ be an isolated $cA_{2}$ point. We identify $X$ with the complete intersection in $\mathbb{P}$ defined by the equations (5.1).

Let $\varphi\colon Y\to X$ be the divisorial contraction of discrepancy $1$ constructed as above. Interchanging the role of $\zeta$ and $\xi$, we obtain another divisorial contraction $\varphi^{\prime}\colon Y^{\prime}\to X$ of discrepancy $1$ centered at $P$. These are all the divisorial contractions of discrepancy $1$ centered at $P$. Thus, the assertion follows from Lemmas 5.4 and 5.5. ∎

Birational rigidity of sextic double solids

Proposition 5.6**.**

Let $X$ be a factorial sextic double solid admitting an ordinary $cA_{2}$ point $P$. Then a divisorial contraction $\varphi\colon Y\to X$ of exceptional type with center $P$ is not a maximal extraction.

Proof.

Let $\varphi\colon Y\to X$ be a divisorial contraction of exceptional type with center a $P$ which is the weighted blow-up with weights $4,3,2,1$ after an appropriate local analytic identification of the germ $P\in X$ by Theorem 2.16. Suppose that it is a maximal extraction. Then there is a mobile linear system $\mathcal{M}\sim_{\mathbb{Q}}-nK_{X}$ such that $(X,\frac{1}{n}\mathcal{M})$ is not canonical at the exceptional divisor $E$ of $\varphi$. Let $D_{1},D_{2}\in\mathcal{M}$ be general members. We can choose a general member $H\in|\mathcal{O}_{X}(1)|$ passing through $P$ such that $H$ does not contain any component of the curve $D_{1}\cap D_{2}$. By Theorem 3.7 and Lemma 2.11, we have

[TABLE]

This is a contradiction and $\varphi$ is not a maximal extraction. ∎

Theorem 5.7**.**

Let $X$ be a factorial sextic double solid with only isolated $cA_{1}$ points and ordinary $cA_{2}$ points. Then $X$ is birationally rigid.

Proof.

By [CP10, Lemma 3.1] and [KOPP24, Propositions 4.8 and 4.9], no smooth point, no curve, and no $cA_{1}$ point on $X$ are maximal centers. Therefore, by Theorem 2.14, the assertion follows from Propositions 5.1 and 5.6. ∎

Fano 3-fold WCIs of index 1

We investigate the birational rigidity of prime Fano 3-folds embedded as a complete intersection in a weighted projective space that admit $cA$ points. By a prime Fano $3$-fold, we mean a normal projective $\mathbb{Q}$-factorial $3$-fold with only terminal singularities such that the anticanonical divisor is ample and the class group is isomorphic to $\mathbb{Z}$. The index of a prime Fano $3$-fold $X$ is defined to be the positive integer $\iota_{X}$ such that $-K_{X}\sim\iota_{X}A$, where $A\in\operatorname{Cl}(X)$ is the ample generator of $\operatorname{Cl}(X)\cong\mathbb{Z}$.

Setting 6.1**.**

Let $X$ be a prime Fano $3$-fold which is either a weighted hypersurface of index $1$ listed in Table LABEL:table:Fanohyp or a weighted complete intersection of codimension $2$ and index $1$ listed in Table 2. Let $\mathbb{P}$ be the ambient weighted projective space of $X$. We assume that $X$ is quasismooth along $X\cap\operatorname{Sing}(\mathbb{P})$, where $\operatorname{Sing}(\mathbb{P})$ is the singular locus of $\mathbb{P}$.

Theorem 6.2**.**

Let $X$ be as in Setting 6.1 and let $k_{\mathrm{cA}}$ be the positive integer given in the 5th and 4th columns of Tables LABEL:table:Fanohyp and 2, respectively. Suppose that $X$ has only $cA_{k}$ points, where $k\leq k_{\mathrm{cA}}$, in addition to terminal cyclic quotient singularities. Then $X$ is birationally rigid.

Basics on weighted projective varieties

We give basic definitions concerning weighted projective spaces and its closed subvarieties. Let $\mathbb{P}=\mathbb{P}(a_{0},\dots,a_{N})$ be a weighted projective space and let $x_{0},\dots,x_{N}$ be its homogeneous coordinates of weights $a_{0},\dots,a_{N}$, respectively. We say that $\mathbb{P}$ is well-formed if the greatest common divisor of any $N$ of the weights $a_{0},\dots,a_{N}$ is $1$. Let $X$ be a closed subvariety of $\mathbb{P}=\mathbb{P}(a_{0},\dots,a_{N})$. We say that $X$ is well-formed if $\mathbb{P}$ is well-formed and

[TABLE]

If $X\subset\mathbb{P}$ is normal and well-formed, then the adjunction holds, that is, we have $(K_{\mathbb{P}}+X)|_{X}=K_{X}$. We define

[TABLE]

where $I$ is the defining homogeneous ideal of $X$ in $\mathbb{P}$, and let $\pi\colon C_{X}^{*}\to X$ be the natural projection, where $C_{X}^{*}:=C_{X}\setminus\{o\}$. We say that $X$ is quasismooth if $C_{X}^{*}$ is smooth. For a subset $S\subset X$, we say that $X$ is quasismooth along $S$ if $C_{X}^{*}$ is smooth along $\pi^{-1}(S)$.

For a homogeneous coordinate $s\in\{x_{0},\dots,x_{N}\}$, we denote by $P_{s}$ the point $(0\!:\!\cdots\!:\!1\!:\!\cdots\!:\!0)\in\mathbb{P}$ at which only the coordinate $s$ does not vanish. For homogeneous polynomials $f_{1},\dots,f_{m}\in\mathbb{C}[x_{0},\dots,x_{N}]$, we define

[TABLE]

to be the closed subscheme of $\mathbb{P}$ defined by the homogeneous ideal $(f_{1},\dots,f_{m})$ of the graded ring $\mathbb{C}[x_{0},\ldots,x_{N}]$.

Definition 6.3**.**

Let $P\in X$ be a point. We say that a set $\{g_{1},\dots,g_{m}\}$ of homogeneous polynomials $g_{1},\dots,g_{m}\in\mathbb{C}[x_{0},\dots,x_{N}]$ isolates $P$ or is a $P$-isolating set if $P$ is an isolated component of the set

[TABLE]

Definition 6.4**.**

Let $P\in X$ be a point that is contained in the smooth locus of $\mathbb{P}$ and let $L$ be a Cartier divisor class on $X$. For positive integers $k$ and $l$, we define $|\mathfrak{m}^{k}(lL)|$ to be the sublinear system of $|lL|$ consisting of divisors vanishing at $P$ with order at least $k$. We say that $L$ isolates $P$ or is a $P$-isolating class if $P$ is an isolated component of the base locus of $|\mathfrak{m}_{P}^{k}(kL)|$ for some positive integer $k$.

Lemma 6.5**.**

Let $P\in X$ be a point that is contained in the smooth locus of $\mathbb{P}$. If a set $\{g_{1},\dots,g_{m}\}$ of homogeneous polynomials $g_{1},\dots,g_{m}\in\mathbb{C}[x_{0},\dots,x_{N}]$ isolates $P\in X$, then $lA$ isolates $P$, where

[TABLE]

Proof.

We set $L:=lA\in\operatorname{Cl}(X)$. We define $n:=\operatorname{lcm}\{\,\deg g_{i}\mid 1\leq i\leq m\,\}$, $k_{i}:=n/\deg g_{i}$ for $1\leq i\leq m$ and $k:=n/l$. Then we have $kL=nA$ and $k_{i}\geq k$ for $1\leq i\leq m$. We have $\operatorname{ord}_{P}(D_{i})\geq 1$ for the effective divisor

[TABLE]

It follows that $k_{i}D_{i}\in|\mathfrak{m}_{P}^{k}(kL)|$ since $\operatorname{ord}_{P}(k_{i}D_{i})\geq k_{i}\geq k$. The base locus of $|\mathfrak{m}_{P}^{k}(kL)|$ is contained in the set $D_{1}\cap\cdots\cap D_{m}$, hence $P$ is its isolated component. This shows that $L=lA$ isolates $P$. ∎

Lemma 6.6**.**

Let $V$ be a normal complete intersection of codimension $c\in\{1,2\}$ in $\mathbb{P}(a_{0},\dots,a_{N})$ and let $A$ be the Weil divisor class of $X$ corresponding to $\mathcal{O}_{V}(1)$.

1. (1)

Let $P\in V$ be a point.

1. (a)

There exists a $P$-isolating class $lA$ for some

[TABLE] 2. (b)

If there is $m$ such that a power of $x_{m}$ appears in one of the defining polynomials of $V$ with nonzero coefficient, then there exist a $P$-isolating class $lA$ for some

[TABLE] 3. (c)

If $c=2$ and there are distinct $m_{1},m_{2}$ such that

[TABLE]

then there exists a $P$-isolating class $lA$ for some

[TABLE] 2. (2)

Let $P\in U_{x_{r}}=(x_{n}\neq 0)\cap V$ be a point, where $r\in\{0,\dots,N\}$.

1. (a)

There exists a $P$-isolating class $lA$ for some

[TABLE] 2. (b)

If there is $m\neq r$ such that a power of $x_{m}$ appears in one of the defining polynomials of $V$ with nonzero coefficient, then there exists a $P$-isolating class $lA$ for some

[TABLE] 3. (c)

If $c=2$ and there are distinct $m_{1},m_{2}$ such that $m_{i}\neq r$ for $i=1,2$ and

[TABLE]

then there exists a $P$-isolating class $lA$ for some

[TABLE]

Proof.

We write $P=(\alpha_{0}\!:\!\cdots\!:\!\alpha_{N})$, where $\alpha_{i}\in\mathbb{C}$. For $i\neq j$, we set $b_{ij}=\operatorname{lcm}(a_{i},a_{j})$ and

[TABLE]

which is a homogeneous polynomial of degree $b_{ij}=\operatorname{lcm}(a_{i},a_{j})$. The set of polynomials

[TABLE]

clearly isolates $P$, which shows (1-a). If $P\in U_{r}=(x_{r}\neq 0)\cap V$, then the set

[TABLE]

isolates $P$.

Suppose that a power of $x_{m}$ appears in one of the defining polynomials of $V$. We consider the projection

[TABLE]

Any fiber of $\pi$ is a finite set of points. Then the set of polynomials

[TABLE]

isolates the point $\pi(P)$. This shows that the above set is a $P$-isolating class as well since $\pi^{-1}(\pi(P))$ is a finite set. This proves (1-b). If in addition $P\in U_{r}$ and $m\neq r$, then $\pi(P)\in(x_{r}\neq 0)\cap\pi(X)$. Since the set

[TABLE]

isolates $\pi(P)$, the same set isolates $P$ as well. This shows (2-b).

Suppose that the assumption of (1-c) is satisfied. We consider the projection

[TABLE]

which is indeed everywhere defined by the assumption (6.1). Let $A^{\prime}\in\operatorname{Cl}(\mathbb{P}^{\prime})$ be the Weil divisor class corresponding to $\mathcal{O}_{\mathbb{P}^{\prime}}(1)$. We have $\pi^{*}A^{\prime}=A$ and this implies that $\pi$ does not contract any curve. The set

[TABLE]

isolates the point $\pi(P)\in\mathbb{P}^{\prime}$ and hence the same set isolates $P$ since $\pi^{-1}(\pi(P))$ is a finite set. This shows (1-c). If in addition $P\in U_{r}$ for some $r\neq m_{1},m_{2}$, then $\pi(P)\in(x_{r}\neq 0)\cap\pi(V)$. In this case the set

[TABLE]

isolates the point $\pi(P)\in\mathbb{P}^{\prime}$ and the same isolates $P$. This shows (2-c) and the proof is complete. ∎

Proof of Theorem 6.2

6.2.1 Weighted hypersurfaces

Let $X=X_{d}\subset\mathbb{P}(a_{0},\dots,a_{4})=:\mathbb{P}$ be a member of family listed in Table LABEL:table:Fanohyp satisfying the assumption of Setting 6.1. We assume that $a_{0}\leq\cdots\leq a_{4}$ and set $A:=-K_{X}\in\operatorname{Cl}(X)$ which is the Weil divisor class corresponding to $\mathcal{O}_{X}(1)$. Let $x,y,z,t,w$ be the homogeneous coordinates of $\mathbb{P}$ of weights $a_{0}=1,a_{1},a_{2},a_{3},a_{4}$, respectively, and we denote by $F=F(x,y,z,t,w)$ the homogeneous polynomial of degree $d$ that defines $X$ in $\mathbb{P}$.

Remark 6.7*.*

We explain the $7$th column “$F(0,0,z,t,w)$” of Table LABEL:table:Fanohyp. The polynomial defining $X$ is denoted by $F=F(x,y,z,t,w)$ and in the $7$th column we give an explicit description of $F(0,0,z,t,w)$ in the following manner:

• •

If $d=2a_{4}$, then may and do assume $F(0,0,z,t,w)=w^{2}+g(z,t)$ for some homogeneous polynomial $g(z,t)$ of degree $d$. This is possible since $w^{2}\in F$ and we can kill the terms divisible by $w$ by replacing $w\mapsto w-h(z,t)$ for a suitable $h$.

• •

A monomial with coefficient $1$ means that it appears in $F(0,0,z,t,w)$ with a nonzero coefficient. If we rescale variables $z,t$ and $w$, then we may assume that the coefficients of these monomials in $F(0,0,z,t,w)$ are all $1$.

• •

A monomial with parentheses means that its coefficient in $F(0,0,z,t,w)$ can be [math].

• •

A pair of monomials with underline means that at least one of them appears in $F$ with nonzero coefficient.

• •

For families №17, 19, 21 and 27, the polynomial $\phi_{(i)}$ is given as follows:

• –

$\phi_{(17)}(t,w)=(w-\alpha_{1}t)(w-\alpha_{2}t)(w-\alpha_{3}t)$.

• –

$\phi_{(19)}(z,t)=(t-\alpha_{1}z)(t-\alpha_{2}z)(t-\alpha_{3}z)(t-\alpha_{4})$.

• –

$\phi_{(21)}(z,t)=(t-\alpha_{1}z^{2})(t-\alpha_{2}z^{2})(t-\alpha_{3}z^{2})$.

• –

$\phi_{(27)}(t,w)=(w-\alpha_{1}t)(w-\alpha_{2}t)(w-\alpha_{3}t)$.

In each of the above equations, the $\alpha_{i}\in\mathbb{C}$ are mutually distinct.

Lemma 6.8**.**

Let $X=X_{d}\subset\mathbb{P}(a_{0},a_{1},\dots,a_{4})$ be a member of a family marked $\clubsuit$ in the $6$th column of Table LABEL:table:Fanohyp and we define

[TABLE]

which is the positive integer given in the $4$th column of Table LABEL:table:Fanohyp. Then, for any point $P\in\operatorname{NQsm}(X)$, there is a $P$-isolating class $lA$ for some $l\leq l_{\mathrm{ic}}$.

Proof.

We have $d=2a_{4}$ and $a_{3}<a_{4}$. This implies $w^{2}\in F$. Then the assertion follows from Lemma 6.6. ∎

Lemma 6.9**.**

Let $X$ be a member of a family marked $\heartsuit$ in the $6$th column of Table LABEL:table:Fanohyp and we define

[TABLE]

Then the non-quasismooth locus $\operatorname{NQsm}(X)$ of $X$ is contained in $U_{x}\cup U_{y}$. For any point $P\in U_{x}\cup U_{y}$, there exists a $P$-isolating class $lA$ for some $l\leq l_{\mathrm{ic}}$.

Proof.

We set $f:=F(0,0,z,t,w)$ and

[TABLE]

Then $(x=y=0)\cap\operatorname{NQsm}(X)\subset\Xi$. The polynomial $f=f(z,t,w)$ is explicitly described in Table LABEL:table:Fanohyp.

Suppose that we are not in one of the following cases:

• (i)

$X$ belongs to family №$32$ and $wz^{3},tz^{4}\notin F$.

• (ii)

$X$ belongs to family №$49$ and $tz^{3}\notin F$.

• (iii)

$X$ belongs to family №$64$ and $tz^{4}\notin F$.

In this case, it is straightforward to check that $\Xi\subset\{P_{z},P_{t},P_{w}\}$. By assumption, $X$ is quasismooth at $P_{z},P_{t},P_{w}$ if $X$ contains $P_{z},P_{t},P_{w}$, respectively. This shows that $X$ is quasismooth along $(x=y=0)\cap X$ and the first assertion is proved in this case.

Suppose that we are in one of the cases (i), (ii) and (iii). Then $X$ belongs to family $64$ and $tz^{4}\notin f$. In this case we have $\Xi=(x=y=t=0)\subset X$ ($\Xi=(x=y=w=0)\subset X$) if we are in the case (i) (resp. (ii) and (iii)), and we can write

[TABLE]

where $\alpha,\beta\in\mathbb{C}$ and $g=g(x,y,z,t,w)$ is a homogeneous polynomial that is contained in the ideal $(x,y,t)^{2}$ (resp. $(x,y,w)^{2}$) if we are in the case (i) (resp. (ii) and (iii)). By the quasismoothness of $X$, we have $\alpha\beta\neq 0$. This implies $(x=y=0)\cap\operatorname{NQsm}(X)=\emptyset$ and thus $X$ is quasismooth along $(x=y=0)\cap X$. This proves the first assertion.

The latter assertion follows from Lemma 6.6. ∎

Lemma 6.10**.**

Let $X$ be a member of a family marked $\spadesuit$ in the $6$th column of Table LABEL:table:Fanohyp and we define

[TABLE]

Then one of the following holds.

1. (1)

The non-quasismooth locus $\operatorname{NQsm}(X)$ of $X$ is contained in $U_{x}\cup U_{y}$. 2. (2)

There is a point $P^{\prime}\in(x=y=0)\cap X$ such that $\operatorname{NQsm}(X)\subset U_{x}\cup U_{y}\cup\{P^{\prime}\}$ and $P^{\prime}\in X$ is an isolated $cA_{1}$ point.

For any point $P\in U_{x}\cup U_{y}$, there exists a $P$-isolating class $lA$ for some $l\leq l_{\mathrm{ic}}$. Moreover, in the case $(2)$, the divisor class $a_{2}a_{3}A$ isolates $P^{\prime}$.

Proof.

The polynomial $f:=F(0,0,z,t,w)$ is explicitly described in the $7$th column of Table LABEL:table:Fanohyp (see also Remark 6.7).

Suppose that at least one of the monomials with parentheses appear in $F$ with a nonzero coefficient. Then we have $\Xi\subset\{P_{z},P_{t},P_{w}\}$, where $\Xi$ is as in (6.2). By the same argument as in the proof of Lemma 6.9, the assertion (1) holds in this case.

Suppose that no monomial with parentheses appears in $f$. Then

[TABLE]

In each instance, we can write

[TABLE]

where and $g=g(x,y,z,t)\in(x,y)^{2}$. We see that $X$ is quasismooth along $\Xi\setminus\{P^{\prime}\}$, where $P^{\prime}:=(0\!:\!0\!:\!1\!:\!1\!:\!0)\in\Xi$, and $X$ is not quasismooth at $P^{\prime}$. Let $\mathcal{U}$ be the open subset $(z\neq 0)\cap(t\neq 0)$ of $\mathbb{P}$ and set $U:=X\cap\mathcal{U}$. We give an explicit description of $U$. We take positive integers $m$ and $n$ such that $ma_{3}-na_{2}=1$ and set $M=t^{m}/z^{n}$. We set $u:=z^{a_{3}}/t^{a_{2}}$, $\tilde{x}=x/M,\tilde{y}=y/M^{a_{1}}$ and $\tilde{w}=w/M^{a_{4}}$. Note that $z/M^{a_{2}}=u^{m}$ and $t/M^{a_{3}}=u^{n}$. Then we have

[TABLE]

and $U$ is the hypersurface defined in $\mathcal{U}$ by the equation

[TABLE]

where $\tilde{g}=g(\tilde{x},\tilde{y},u^{m},u^{n})\in(\tilde{x},\tilde{y})^{2}$. The ponit $P^{\prime}\in U$ corresponds to the point $(\tilde{x},\tilde{y},\tilde{w},u)=(0,0,0,1)$. We set $\tilde{u}:=u-1$ and we make the coordinate change $u\mapsto\tilde{u}$. Then

[TABLE]

where $\tilde{g}^{\prime}=\tilde{g}(\tilde{x},\tilde{y},\tilde{u})\in(\tilde{x},\tilde{y})^{2}$. The point $P^{\prime}$ corresponds to the origin. The quadratic part of $\tilde{F}^{\prime}$ contains the monomials $\tilde{y}\tilde{u}$ and $\tilde{w}^{2}$ and it is of rank at least $3$. A Gorenstein terminal singularity is a hypersurface singularity, and it is of type $cA_{k}$ for some $k\geq 1$ (resp. $cA_{1}$) if and only if the quadratic part of its defining equation is of rank at least $2$ (resp. at least $3$). It follows that $P^{\prime}\in X$ is an isolated $cA_{1}$ point. Thus we are in case (2).

The existence of a $P$-isolating divisor for $P\in U_{x}\cup U_{y}$ follows from Lemma 6.6. In the case (2), the set of polynomials $\{x,y,z^{a_{3}}-t^{a_{2}},w\}$ clearly isolates the point $P^{\prime}\in X$, and hence $a_{2}a_{3}A$ isolates $P^{\prime}$ since $a_{2}a_{3}>a_{4}$. ∎

For a family listed in Table LABEL:table:Fanohyp, the positive integer $k_{\mathrm{cA}}$ in the $5$th column is defined by

[TABLE]

where, for a real number $r$, $\lfloor r\rfloor$ is the greatest integer which is less than or equal to $r$.

Proposition 6.11**.**

Let $X$ be a member of a family listed in Table LABEL:table:Fanohyp satisfying the assumption of Setting 6.1 and let $k_{\mathrm{cA}}$ be the positive integer in the $5$th column of Table LABEL:table:Fanohyp. Then no $cA_{k}$ point is a maximal center for $k\leq k_{\mathrm{cA}}$.

Proof.

If $X$ belongs to a family marked $\bigstar$, then the assertion follows from [KOPP24, Proposotion 4.9] and we do not treat these families.

Suppose that $P\in X$ is a $cA_{k}$ point that is a maximal center, where $k\leq k_{\mathrm{cA}}$. Then there exist a positive integer $n$ and a mobile linear system $\mathcal{M}\subset\left|-nK_{X}\right|$ such that $P$ is a center of non-canonical singularities of the pair $(X,\frac{1}{n}\mathcal{M})$. Let $D_{1},D_{2}\in\mathcal{M}$ be general members. If $X$ belong to a family marked $\spadesuit$ and $P$ is contained in $(x=y=0)\cap X$, then $k=1$ and we have

[TABLE]

by Theorem 3.8. Otherwise we have

[TABLE]

By Lemmas 6.8, 6.9 and 6.10, there is a $P$-isolating class $a_{2}a_{3}A$ (resp. $lA$ for some $l\leq l_{\mathrm{ic}}$) if $X$ belongs to a family marked $\spadesuit$ and $P\in(x=y=0)\cap X$ (resp. otherwise). It follows that there is an effective $\mathbb{Q}$-divisor $T\sim_{\mathbb{Q}}-a_{2}a_{3}K_{X}$ (resp. $T\sim_{\mathbb{Q}}-lK_{X}$) such that $\operatorname{ord}_{P}(T)\geq 1$ and that $\operatorname{Supp}(T)$ does not contain any component of $D_{1}\cdot D_{2}$ passing through $P$ if $X$ belongs to a family marked $\spadesuit$ and $P\in(x=y=0)\cap X$ (resp. otherwise). If $X$ belongs to a family marked $\spadesuit$ and $P\in(x=y=0)\cap X$, then by Lemma 2.11 and (6.5) we have

[TABLE]

which is impossible since $a_{2}a_{3}(-K_{X}^{3})\leq 2$ in each instance. Otherwise, by Lemma 2.11 and (6.6), we have

[TABLE]

which is impossible by (6.4). Thus $P\in X$ cannot be a maximal center. ∎

6.2.2 Weighted complete intersections of codimension 2

Let $X=X_{d_{1},d_{2}}\subset\mathbb{P}(a_{0},\dots,a_{5})=:\mathbb{P}$ be a member of family listed in Table 2 satisfying the assumption of Setting 6.1. We assume that $a_{0}\leq\cdots\leq a_{5}$ and set $A:=-K_{X}\in\operatorname{Cl}(X)$ which is the Weil divisor class corresponding to $\mathcal{O}_{X}(1)$. Let $x,y,z,t,v,w$ be the homogeneous coordinates of $\mathbb{P}$ of weights $a_{0},a_{1},a_{2},a_{3},a_{4},a_{5}$, respectively, and we denote by $F_{1}=F_{1}(x,y,z,t,v,w)$ and $F_{2}=F_{2}(x,y,z,t,v,w)$ the homogeneous polynomials of degree $d_{1}$ and $d_{2}$, respectively, that define $X$ in $\mathbb{P}$.

Lemma 6.12**.**

Let $X$ be a member of a family marked $\clubsuit_{m,5}$ for some $m\in\{3,4\}$ in the $6$th column of Table 2 and we define

[TABLE]

Then, for any point $P\in\operatorname{NQsm}(X)$, there is a $P$-isolating class $lA$ for some $l\leq l_{\mathrm{ic}}$.

Proof.

We set $s:=t$ (resp. $s:=v$) if $m=3$ (resp. $m=4$. Then it is easy to see that

[TABLE]

The assertion follows by applying (1-c) of Lemma 6.6 for $(m_{1},m_{2})=(m,5)$. ∎

Lemma 6.13**.**

Let $X$ be a member of a family marked $\heartsuit_{m,5}$ for some $m\in\{3,4\}$ in the 6th column of Table 2 and we define

[TABLE]

Then the non-quasismooth locus $\operatorname{NQsm}(X)$ of $X$ is contained in $U_{x}\cup U_{y}$. For any point $P\in U_{x}\cup U_{y}$, there exists a $P$-isolating class $lA$ for some $l\leq l_{\mathrm{ic}}$.

Proof.

We consider the matrix

[TABLE]

and then define

[TABLE]

We see that $\operatorname{NQsm}(X)\cap(x=y=0)\subset\Xi$.

By a straightforward computation in each instance, we can easily compute the set $\Xi$ and it is described in the 5th column of Table 3.

Suppose that $X$ belongs to the family №$71$ and $tz^{2}\notin F_{2}$. In this case $\Xi=(x=y=w=0)$. We have $z^{3}x\in F_{2}$ (see Remark 6.14) and we may assume that its coefficient is $1$ by rescaling $x$. Then we can write

[TABLE]

where $\alpha,\beta\in\mathbb{C}$ and $g_{i}=g_{i}(x,y,z,t,w)$ is a homogeneous polynomials that is contained in the ideal $(x,y,v,w)^{2}$. By the quasismoothness of $X$ at $P_{z}$ and $P_{t}$, we have $\alpha\neq 0$ and $\beta\neq 0$. Then we see that the Jacob matrix of $X$ is of rank $2$ along the set $(z\neq 0)\cap(x=y=0)\cap X$. It follows that $\operatorname{NQsm}(X)\cap(x=y=0)\subset\{P_{t}\}$.

In the other instances, we have $\Xi\subset X\cap\operatorname{Sing}(\mathbb{P})$, and hence we have $\operatorname{NQsm}(X)\subset X\cap\operatorname{Sing}(\mathbb{P})$ in all instances. This proves the first assertion since $X$ is quasismooth along $X\cap\operatorname{Sing}(\mathbb{P})$.

We set $s:=t$ (resp. $s:=v$) if $m=3$ (resp. $m=4$). Then it is easy to see that

[TABLE]

The second assertion follows by applying (2-c) of Lemma 6.6 for $(m_{1},m_{2})=(m,5)$. ∎

Remark 6.14*.*

We explain the equations given in Table 3 in more detail. In the $2$nd and $3$rd columns we give an explicit descriptions of $F_{1}(0,0,z,t,v,w)$ and $F_{2}(0,0,z,t,v,w)$, respectively, in the following manner:

• •

We have $d_{2}=2a_{5}$ and we have $w^{2}\in F_{2}$ by the quasismoothness assumption. We may and do assume that $F(0,0,z,t,v,w)=w^{2}+g(z,t,v)$ for some homogeneous polynomial $g(z,t,v)$ of degree $d_{2}$. This is possible since we can kill the terms divisible by $w$ by replacing $w\mapsto w-h(z,t,v)$ for a suitable $h$.

• •

For family №$71$, we have case divisions as follows: We are in the first case if $tz^{2}\in F_{1}$, otherwise we are in the second case. In the second case, we have $z^{3}x\in F_{2}$ by the quasismoothness assumption.

• •

If we rescale variables $z,t$ and $w$, then we may assume that the coefficients of these monomials in $F(0,0,z,t,w)$ are all $1$.

For a family listed in Table 2, the positive integer $k_{\mathrm{cA}}$ in the $5$th column is defined by

[TABLE]

Proposition 6.15**.**

Let $X$ be a member of a family listed in Table 2 satisfying the assumption of Setting 6.1 and let $k_{\mathrm{cA}}$ be the positive integer in the $5$th column of Table 2. Then no $cA_{k}$ point is a maximal center for $k\leq k_{\mathrm{cA}}$.

Proof.

The proof is identical to that of Proposition 6.11. If $X$ belongs to a family marked $\bigstar$, then the assertion follows from [KOPP24, Proposotion 4.9] and we do not treat these families.

Suppose that $P\in X$ is a $cA_{k}$ point that is a maximal center, where $k\leq k_{\mathrm{cA}}$. Then there exist a positive integer $n$ and a mobile linear system $\mathcal{M}\subset\left|-nK_{X}\right|$ such that $P$ is a center of non-canonical singularities of the pair $(X,\frac{1}{n}\mathcal{M})$. Let $D_{1},D_{2}\in\mathcal{M}$ be general members. By Lemmas 6.12, and 6.13, there is a $P$-isolating class $lA$ for some $l\leq l_{\mathrm{ic}}$. It follows that there is an effective $\mathbb{Q}$-divisor $T\sim_{\mathbb{Q}}-lK_{X}$ such that $\operatorname{ord}_{P}(T)\geq 1$ and that $\operatorname{Supp}(T)$ does not contain any component of $D_{1}\cdot D_{2}$ passing through $P$. We have

[TABLE]

where the second inequality follows from Lemma 2.11 and the third inequality follows from Theorem 3.8. This is impossible by (6.7). Thus $P\in X$ cannot be a maximal center. ∎

6.2.3 Completion of Proof of Theorem 6.2

Let $X$ be as in Theorem 6.2. Then, by [KOPP24, Propositions 5.5 and 5.12], no curve and no smooth point is a maximal center of $X$. Moreover, by [KOPP24, Proposition 5.15], for each terminal quotient singular point $P\in X$, either it is not a maximal center, or there is an elementary self-link initiated by the Kawamata blow-up of $X$ at $P$. Note that these results are due to [CP17] as it is explained in [KOPP24]. Finally, by Proposition 6.11 and 6.15, no $cA_{k}$ point is a maximal center for $k\leq k_{\mathrm{cA}}$. By Theorem 2.14, $X$ is birationally rigid and Theorem 6.2 is proved.