On the number of singular points of factorial terminal Fano threefolds

TL;DR

This paper establishes upper bounds on the number of singular points that factorial terminal Fano threefolds can have, contributing to the classification and understanding of their singularities.

Contribution

It provides new upper bounds on singular points for factorial terminal Fano threefolds, advancing the classification of these varieties.

Findings

Upper bounds on singular points established

Enhanced understanding of factorial terminal Fano threefolds

Contributes to classification theory

Abstract

We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.

Equations

$$s(X)\le 20-\mathrm{rk}\mathrm{Pic}(X)+h^{1,2}({\mathfrak{X}}_s),$$

$$Y\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$s(X)\le s(Y)+1=s(Y^{\prime })+1=s(Z)+s(\Gamma )+1\le s(Z)+p_a(\Gamma )+1$$

$$\mathrm{deg}\Lambda =\frac{1}{2}D\cdot E\cdot (-K_Y)=7\alpha \ge 7$$

$$s(X)=s(Y)=s(Y^{\prime })=s(Z)+s(\Gamma )\le s(Z)+p_a(\Gamma ).$$

Full text

This work is supported by the Russian Science Foundation under grant 14-50-00005.

1. Introduction

Throughout this paper by $X$ we denote a Fano threefold with terminal ${\mathbb{Q}}$-factorial singularities and $\operatorname{rk}\operatorname{Pic}(X)=1$. Let $\iota=\iota(X)$ be its Fano index and $g:=(-K_{X})^{3}/2+1$ be its genus. When $\iota(X)\geq 2$, more convenient invariant of a Fano threefold $X$ is its degree $d:=(-K_{X})^{3}/\iota^{3}$. Recall that any $X$ as above has a one-parameter smoothing $\mathfrak{X}$ [7]. Let $\operatorname{h}^{1,2}(\mathfrak{X}_{s})$ be the Hodge number of a general fiber of $\mathfrak{X}$. Denote by $s(V)$ be the number of singular points of $V$.

The upper bounds of $s(X)$ are interesting for classification problems, in particular, for the classification of finite subgroups of the space Cremona group (see., e.g., [8], [10], [11], [14]). Y. Namikawa in [7] proved the inequality

[TABLE]

which holds for any Fano threefold $X$ with terminal Gorenstein singularities. However, this estimate is quite rough. For Fano threefolds with non-degenerate singularities it is known the inequality $s(X)\leq\operatorname{h}^{1,2}(\mathfrak{X}_{s})$ (see, e.g., [12, §10]).

Theorem 1** ((cf. [9, Thm. 4.1])).**

Suppose either 1) $\iota=1$ and $g\geq 7$ or 2) $\iota=2$ and $d\geq 3$. Then $s(X)\leq\operatorname{h}^{1,2}(\mathfrak{X}_{s})$ and this bound is attained for some $X$ having only ordinary double points.

Recall (see [4], see also the references in [13]) that $\operatorname{h}^{1,2}(\mathfrak{X}_{s})$ takes the following values:

[TABLE]

[TABLE]

Note that in our theorem we do not assume that the singularities of $X$ are non-degenerate. Moreover, our construction allows to get better bounds in degenerate cases. If $\iota=1$, $g=6$ or $\iota=d=2$, then under the additional assumption that the variety $X$ has a $cA_{1}$-point, our computations allow us to get the estimate $s(X)\leq 15$ (cf. [9, proof of 1.3]). However, we do not claim that this bound is sharp.

A Sarkisov link is the following diagram

[TABLE]

where $f$ and $f^{\prime}$ are extremal Mori contractions and $\chi$ is a flop. In our situation, the morphism $f$ is birational. Let $E$ be the exceptional divisor and $E^{\prime}\subset Y^{\prime}$ be its proper transform. If $f^{\prime}$ is also birational, then we denote by $D^{\prime}$ its exceptional divisor and $\Gamma:=f^{\prime}(D^{\prime})$ If, moreover, $\Gamma$ is curve, then it is irreducible, contained in the non-singular locus of $Z$, $f^{\prime}$ is the blowup of $\Gamma$ and the singularities $\Gamma$ are planar (if there are any, see [2]).

2. Birational transformations in singular points.

Recall that a three-dimensional singularity $P\in X$ is of type $cA_{1}^{m}$, if it is analytically equivalent to a hypersurface singularity $x_{1}x_{2}+x_{3}^{2}+x_{4}^{m}$. The blowup $f:Y\to X$ of the the maximal ideal $\mathfrak{m}_{P,X}$ of such a point is a Mori contraction (see [2]). If the $m\leq 2$, then $Y$ is smooth along $f^{-1}(P)$. If $m>2$, then $Y$ has on $f^{-1}(P)$ exactly one singular point and this point is of type $cA_{1}^{m-2}$.

Sarkisov links with centers at singular points $P\in X$ of type $cA_{1}$ appeared earlier in different papers (see, e.g., [3], [5], [1, §3], [15]) but usually under some additional restrictions. My systematize and generalize this information:

Theorem 2**.**

Suppose that $X$ has a $cA_{1}$-point $P\in X$. Let $f:Y\to X$ be the blowup of $\mathfrak{m}_{P,X}$. If $\iota=2$, $d\geq 2$, then $f$ is included in the Sarkisov link (1), where $\chi$ is an isomorphism and $f^{\prime}$ is defined by the linear system $|\frac{1}{2}(-K_{Y^{\prime}}-E^{\prime})|$. Except for the case 2, the morphism $f^{\prime}$ contracts a divisor $D^{\prime}$ to a curve $\Gamma$:

[TABLE]

If $\iota=1$, $g\geq 4$, then $f$ is included in the Sarkisov link (1), where, except for the case 2, the morphism $f^{\prime}$ is defined by the linear system $|-K_{Y^{\prime}}-E^{\prime}|$, and, except the cases 2 and 2, $f^{\prime}$ contracts a divisor $D^{\prime}$ to a curve $\Gamma$:

[TABLE]

In the case 2, $D^{\prime}\in|-K_{Y^{\prime}}-E^{\prime}|$.

Proof.

In all our cases, the linear system $|-K_{X}|$ is very ample and defines an embedding $X\subset{\mathbb{P}}^{g+1}$ as a variety of degree $2g-2$. Consider the projection $\psi:X\dashrightarrow Y_{0}\subset{\mathbb{P}}^{g}$ from $P$. The blowup $f:Y\to X$ of the maximal ideal $\mathfrak{m}_{P,X}$ resolves the indeterminacy points of $\psi$ and gives a morphism $Y\to Y_{0}$. Note that the exceptional divisor $E\subset Y$ is isomorphic to an irreducible quadric $Q\subset{\mathbb{P}}^{3}$ and ${\mathscr{O}}_{E}(E)\simeq{\mathscr{O}}_{Q}(-1)$ (see [2]). It is easy to see that $K_{Y}=f^{*}K_{X}+E$. This shows that the morphism $Y\to Y_{0}$ is given by the anticanonical linear system and $-K_{Y}^{3}=-K_{X}^{3}-2>0$. Therefore, $\dim Y_{0}=3$ and in the Stein factorization $Y\to\bar{Y}\to Y_{0}$, the morphism $\bar{f}:Y\to\bar{Y}$ is birational and crepant. Since there are at most a finite number of lines passing through $P$, the exceptional locus of the projection $\psi$ is at most one-dimensional. Thus, $\bar{f}:Y\to\bar{Y}$ is a small crepant contraction. Hence there exists a flop $\chi:Y\dashrightarrow Y^{\prime}$ and an extremal Mori contractions $f^{\prime}:Y^{\prime}\to Z$. Using the restriction exact sequences to $E\simeq Q\subset{\mathbb{P}}^{3}$, it is easy to compute that $\dim|-K_{Y^{\prime}}-E^{\prime}|\geq\dim|-K_{X}|=g-4$ always and $|\frac{1}{2}(-K_{Y^{\prime}}-E^{\prime})|\geq d$ for $\iota=2$. Using this and solving the corresponding Diophantine equations, as in [4, ch. 4] and [9, §4], we obtain the possibilities 2-2. The smoothness of $Z$ in the cases 2, 2 and 2 follows from ${\mathbb{Q}}$-factoriality (see Corollary 2 below). If $\iota=2$, then the threefold $X$ does not contain lines (for the anticanonical embedding). Therefore, the divisor $-K_{Y}$ is ample and $Y\simeq Y^{\prime}$. This proves Theorem 2. ∎

Remark 1**.**

Let the pair $(Z,\Gamma)$ be such as in the cases 2, 2, 2-2. Suppose that

1. (i)

the curve $\Gamma$ has planar singularities and is contained in the non-singular locus of $Z$, 2. (ii)

$\Gamma$* is a scheme-theoretic intersection of elements of the linear system $|\iota(Z)L|$, where $L$ is the ample generator of $\operatorname{Pic}(Z)\simeq{\mathbb{Z}}$,* 3. (iii)

the number of $\iota(Z)$-secant lines of $\Gamma$ is at most finite.

Then the blowup of $\Gamma$ can be completed to a Sarkisov link with the corresponding Fano threefold $X$ (i.e. the construction (1) can be reversed). Indeed, it follows from the condition (i) that $Y^{\prime}$ has only terminal factorial singularities, (ii) that $|-K_{Y^{\prime}}|$ has no base points, and (iii) guarantees that the corresponding morphism defines a crepant small contraction. The rest is similar to the proof of Theorem 2.

Corollary 1**.**

Suppose that in the conditions of Theorem 1, the threefold $X$ has a point of type $cA_{1}$. Then $s(X)\leq\operatorname{h}^{1,2}(\mathfrak{X}_{s})$ and this bound is attained for some $X$ having only ordinary double points.

Proof.

Note that three-dimensional terminal flops preserve types of singularities. Therefore, in the notation of Theorem 2 and (1) we have

[TABLE]

(see [9, 5.1(ii)]). In all cases 2-2 we can choose $Z$ and $\Gamma$ so that equalities are attained. This proves Corollary 1. ∎

3. Proof of Theorem 1: the case $g\neq 7$

For $\iota=1$, $g\geq 9$, Theorem 1 was proved in [9, 1.3], and for $\iota=1$, $g=8$ and $\iota=2$, $d=3$ the assertion follows directly from Corollary 1 and the following lemma.

Lemma 1**.**

Assume either: 1) $\iota=1$ and $g=8$, or 2) $\iota=2$ and $d=3$. If $s(X)>2$, then $X$ has a singularity of type $cA_{1}$.

Proof.

In the case $g=8$, according to [9, 4.1], the double projection from a line contained in the non-singular locus determines a Sarkisov link, where $f^{\prime}$ is a conic bundle over ${\mathbb{P}}^{2}$ with discriminant curve $\Delta$ of degree $5$. Similarly, for a cubic hypersurface with ${\mathbb{Q}}$-factorial terminal singularities the usual projection from a line defines a Sarkisov link, where $f^{\prime}$ is conic bundle over ${\mathbb{P}}^{2}$ with discriminant curve $\Delta$ degree $5$ (in this case $\chi$ is isomorphism). Moreover, $Y^{\prime}$ has the same collection of singularities as $X$. Assume that $s(X)>2$ and all the singularities of $X$ are worse than $cA_{1}$. Let $R\in Y^{\prime}$ be a singular point and $o:=f^{\prime}(R)$.

We claim that $R$ is the only singular point of $Y^{\prime}$ lying on the fiber $f:={f^{\prime}}^{-1}(o)$ and the singularity of the curve $\Delta$ at $o$ is of multiplicity $\geq 3$. Let $U\subset Z={\mathbb{P}}^{2}$ be a small analytic neighborhood of $o$ and $Y^{\prime}_{U}:={f^{\prime}}^{-1}(U)$. Then $Y_{U}^{\prime}$ can be embedded to ${\mathbb{P}}^{2}_{x_{0},x_{1},x_{2}}\times U_{u_{1},u_{2}}$ and defined there by a homogeneous equation of degree $2$ with respect to the variables $x_{0},x_{1},x_{2}$. We may assume that $R$ has coordinates $(0:0:1;0,0)$.

If $F$ is a reducible conic, then the equation of $Y^{\prime}_{U}$ can be written in the form $x_{0}x_{1}+\alpha(u_{1},u_{2})x_{2}^{2}=0$, where $\alpha\in\mathfrak{m}_{o,U}^{3}$. In this case $R$ is the only singular point of $Y^{\prime}_{U}$ and the discriminant curve $\Delta=\{\alpha=0\}$ has at $o$ a singularity of multiplicity $\geq 3$.

If $F$ is a double line, then the equation of $Y^{\prime}_{U}$ can be written in the form $x_{0}^{2}+\alpha x_{1}^{2}+\beta x_{1}x_{2}+\gamma x_{2}^{2}=0$, where $\alpha\in\mathfrak{m}_{o,U}$, $\beta,\gamma\in\mathfrak{m}_{o,U}^{2}$. Since singularities of $Y^{\prime}_{U}$ are isolated, we have $\alpha\notin\mathfrak{m}_{o,U}^{2}$. This means that $R$ is the only singular point of $Y^{\prime}_{U}$. The discriminant curve $\Delta$ is given by $\beta^{2}=4\alpha\gamma$ and again has at $o$ a singularity of multiplicity $\geq 3$. This proves our assertion.

Thus, $s(Y^{\prime})$ is less or equal to the number of points of multiplicity $\geq 3$ of the curve $\Delta$. On the other hand, a plane (possibly reducible) curve of degree 5 has at most two triple points. Lemma 1 is proved. ∎

It remains to consider the case $\iota=2$, $d\geq 4$. The following lemma is well known in the non-singular case (see, e.g., [4]). A singular case is similar.

Lemma 2**.**

Let $X=X_{d}\subset{\mathbb{P}}^{d+1}$ be a Fano threefold with $\iota=2$, $d\geq 4$ and let $l$ be a line contained in the non-singular locus of $X$. Then the projection from $l$ determines a Sarkisov link, where $\chi$ is an isomorphism. Moreover,

1. (i)

if $d=4$, then $Z\simeq{\mathbb{P}}^{3}$, and $f^{\prime}$ is the blowup of an (irreducible) curve of degree $5$ and arithmetic genus $2$ lying on a quadric; 2. (ii)

if $d=5$, then $Z$ is smooth quadric in ${\mathbb{P}}^{4}$ and $f^{\prime}$ is the blowup of a smooth rational cubic curve lying on a hyperplane section.

Corollary 2**.**

If $\iota=2$, $d=5$, then $X$ is smooth. Let $\iota=2$, $d=4$. Then $s(X)\leq 2$ and if $s(X)=2$, then both singularities of $X$ are of type $cA_{1}$.

4. The case $g=7$

Theorem 3**.**

Let $\iota=1$, $g=7$ and let $C\subset X$ be a sufficiently general conic. Then there exists a Sarkisov link (1) with center $C$ and there are two possibilities:

1. (i)

$Z\subset{\mathbb{P}}^{4}$* is a smooth quadric, $f^{\prime}$ is the blowup of a curve $\Gamma\subset Z$ with $\deg\Gamma=10$, $p_{a}(\Gamma)=7$.* 2. (ii)

$Z\subset{\mathbb{P}}^{4}$* is a cubic and $f^{\prime}$ is the blowup of a smooth rational curve $\Gamma\subset Z$ of degree $4$.*

The proof substantially is similar to [4, §4.4] but in the singular case some of the arguments must be modified. According to [7] and [4, 4.2.5] the family $\mathscr{C}$ parametrizing non-degenerate conics on $X$ is not empty and two-dimensional, and according to [6, A.1.2] conics from the family $\mathscr{C}$ cover an open subset in $X$. Further, you need the following.

Lemma 3**.**

1. (i)

For each line $l\subset X$ there are only a finite number of lines on $X$ meeting $l$. 2. (ii)

Let $l$ be a line contained in the non-singular locus of $X$. Then it meets a one-dimensional family of conics. 3. (iii)

There is a non-degenerate conic contained in the non-singular locus of $X$.

Proof.

(i) follows from the fact that $\dim|H-2l|\geq 1$ and $\operatorname{Pic}(X)={\mathbb{Z}}\cdot[H]$.

(ii) Consider the double projection from $l$ and the corresponding Sarkisov link [9]. In the notation of (1), the morphism $f$ is the blowup of $l$, $Z\simeq{\mathbb{P}}^{1}$ and $f^{\prime}$ is a del Pezzo fibration of degree 5. The proper transform $C^{\prime}\subset Y^{\prime}$ of a conic $C\subset X$ meeting $l$ is a line in a fiber of $f^{\prime}$ (because $-K_{Y^{\prime}}\cdot C^{\prime}=1$ and $E^{\prime}\cdot C^{\prime}\geq 1$). There is at most a one-dimensional family of such lines. This proves (ii).

(iii) Assume that all of the conics from the (two-dimensional) family $\mathscr{C}$ pass through a point $P\in X$. Take line $l\subset X$ that does not pass through $P$ and all the lines $l_{i}$ meeting $l$ also do not pass through $P$. Then the map $\chi\circ f^{-1}:X\dashrightarrow Y^{\prime}$ is an isomorphism near $P$. The line $l$ intersects a one-dimensional family of conics from the family $\mathscr{C}$ and proper transforms these conics on $Y^{\prime}$ are lines in the fiber of $f^{\prime}$ passing through the point $P^{\prime}:=\chi(f^{-1}(P))$. Hence this fiber is a cone in ${\mathbb{P}}^{5}$ with vertex $P^{\prime}$. But since the singularities $Y^{\prime}$ are hypersurface, the dimension of the tangent space to a fiber of $f^{\prime}$ is at most 4. The contradiction proves our lemma. ∎

Thus there exists a conic $C$ contained in the non-singular locus of $X$. Take such a conic so that it is not contained in the surface swept out by lines. Consider the projection of $\psi:X\dashrightarrow Y_{0}\subset{\mathbb{P}}^{5}$ from the linear span of $C$. Since $X\cap\langle C\rangle=C$, the blowup $f:Y\to X$ of the conic $C$ resolves the indeterminacy points of $\psi$ and defines a morphism $f_{0}:Y\to Y_{0}$. Here in the Stein factorization $Y\to\bar{Y}\to Y_{0}$, the morphism $\bar{f}:Y\to\bar{Y}$ is birational and is given by a multiple of the linear system $|-K_{Y}|$, i.e., $\bar{f}$ is crepant.

Lemma 4**.**

The exceptional locus of $\bar{f}$ is one-dimensional.

Proof.

Suppose that $\bar{f}$ contracts a divisor $D$. It is easy to compute (see [4, 4.1.2]) that $D\sim\alpha(2(-K_{Y})-3E)$ for some $\alpha\in{\mathbb{Z}}_{>0}$ and $D\cdot E\cdot(-K_{Y})=14\alpha>0$. In particular, $\Lambda:=f_{0}(D\cap E)$ is a curve. Since the conic $C$ meets only a finite number of lines, a general fiber of $D$ must be a conic meeting $C$ at two points [4, 4.4.1]. If the restriction $f_{0}|_{E}:E\to f_{0}(E)$ is a birational morphism, $f_{0}(E)$ is a ruled surface of degree 4 singular along $\Lambda$, where

[TABLE]

This is impossible, see e.g. [4, 4.4.8]. Therefore $f_{0}$ is a morphism of degree 2, $f_{0}(E)$ is a quadric, and $X_{0}\subset{\mathbb{P}}^{5}$ is a subvariety of degree 3. Since $\operatorname{rk}\operatorname{Cl}(\bar{Y})=\operatorname{rk}\operatorname{Cl}(Y_{0})=1$, the variety $Y_{0}$ is a cone over a rational twisted cubic curve with a vertex in a line. However, this cone contains no quadrics. The contradiction proves Lemma 4. ∎

Further, as in [4, §4.1], there exists a flop $\chi:Y\dashrightarrow Y^{\prime}$ and an extremal Mori contraction $f^{\prime}:Y^{\prime}\to Z$. By solving the corresponding Diophantine equations, as in [4, ch. 4] or [9, §4], we obtain the possibilities (i)-(ii). Theorem 3 is proved. For the proof of Theorem 1 in the case $g=7$, similar to (2), we write

[TABLE]

If the possibility (i) of Theorem 3 occurs, then the variety $Z$ is smooth and therefore $s(X)\leq p_{a}(\Gamma)=7$. Otherwise the curve $\Gamma$ is smooth and therefore $s(X)=s(Z)$. Then $s(Z)\leq 5$ according to the already proved case of Theorem 1.