Some examples of Calabi-Yau pairs with maximal intersection and no toric model

TL;DR

This paper presents examples of Calabi-Yau pairs with maximal intersection that cannot be modeled torically, highlighting differences from the surface case and expanding understanding of higher-dimensional Calabi-Yau geometry.

Contribution

It provides the first known examples of higher-dimensional Calabi-Yau pairs with maximal intersection that lack a toric model, contrasting with the surface case.

Findings

Existence of Calabi-Yau pairs with no toric model in higher dimensions

Maximal intersection does not imply toric representability in higher dimensions

Examples challenge assumptions based on surface case behavior

Abstract

It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that admit no toric model.

Tables

Table 1. Table 1. Dimension 2 2 2 slc hypersurface singularities

type	name	symbol	equation $f\in ℂ[x,y,z]$		${\mathrm{mult}}_{0}f$
terminal	smooth	$A_0$	$x$		1
canonical	du Val	$A_n$	$x^2+y^2+z^{n+1}$	$n\ge 1$	2
		$D_n$	$x^2+z\left(y^2+z^{n-2}\right)$	$n\ge 4$	2
		$E_6$	$x^2+y^3+z^4$		2
		$E_7$	$x^2+y^3+y{z}^3$		2
		$E_8$	$x^2+y^3+z^5$		2
lc	simple elliptic	$X_{1,0}$	$x^2+y^4+z^4+\lambda xyz$	${\lambda }^4\ne 64$	2
		$J_{2,0}$	$x^2+y^3+z^6+\lambda xyz$	${\lambda }^6\ne 432$	2
		$T_{3,3,3}$	$x^3+y^3+z^3+\lambda xyz$	${\lambda }^3\ne -27$	3
	cusp	$T_{p,q,r}$	$x^p+y^q+z^r+xyz$		2 or 3
slc	normal crossing	$A_{\infty }$	$x^2+y^2$		2
	pinch point	$D_{\infty }$	$x^2+y^{2}z$		2
	degenerate cusp	$T_{2,\infty ,\infty }$	$x^2+y^2+z^2$		2
		$T_{2,q,\infty }$	$x^2+y^2\left(z^2+y^{q-2}\right)$	$q\ge 3$	2
		$T_{\infty ,\infty ,\infty }$	$xyz$		3
		$T_{p,\infty ,\infty }$	$xyz+x^p$	$p\ge 3$	3
		$T_{p,q,\infty }$	$xyz+x^p+y^q$	$q\ge p\ge 3$	3

Equations

$$U^{\mathrm{trop}}(\mathbb{Z})=\Big{\{}\mbox{divisorial valuations }v\colon k(U)\setminus\{0\}\to\mathbb{Z}\mbox{ with }v(\Omega)<0\Big{\}}\cup\{0\}.$$

$$p:\mathrm{Spec}(V)\to \mathrm{Spec}(R)=T_{\mathrm{Pic}(Y)}$$

$$\mathrm{Spec}(V)/T_K\to T_{\mathrm{Pic}(U)}$$

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

$$K_{W}+\operatorname{Diff}_{W}D_{\widetilde{X}}\sim_{\mathbb{Q}}\big{(}K_{\widetilde{X}}+D_{\widetilde{X}}\big{)}_{|W}.$$

$$D_{\tilde{X}}=\sum _i{f}_{\ast }^{-1}{D}_i+E,$$

$$K_{{\tilde{D}}_i}=(K_{\tilde{X}}+D_i{)}_{|{\tilde{D}}_i}=(f_{|{\tilde{D}}_i}{)}^{\ast }{K}_{D_i}-(E{)}_{|{\tilde{D}}_i}$$

$$f_i:{\tilde{D}}_i\stackrel{q}{\to }{\overline{D}}_i\stackrel{\mu }{\to }{D}_i$$

$$X=\{x_1^4+x_2^4+x_3^4+x_0{x}_1{x}_2{x}_3+x_4(x_0^3+x_4^3)=0\}$$

$$\{0\}\in \{x^4+y^4+z^4+xyz=0\}.$$

$$D_X⇢{\mathbb{P}}_{x_1,x_2,x_3}^2;$$

$$K_{X_p}+D+E=f^{\ast }(K_X+D).$$

$$\left(\begin{array}[]{cccccccc}u&x_{0}&s_{1}&s_{2}&s_{3}&s_{4}\\ 1&0&-1&-1&-1&-1\\ 0&1&1&1&1&1\end{array}\right).$$

$$X_{p}=\{u^{2}\big{(}u(s_{1}^{4}+s_{2}^{4}+s_{3}^{4}+(x_{0}s_{1}s_{2}s_{3})\big{)}+s_{4}(x_{0}^{3}+u^{3}s_{4}^{3})=0\},$$

$$C_1=\{u=s_1=s_4=0\}\subset X_p.$$

$${\mathcal{N}}_{C_1/X_p}\simeq {\mathcal{N}}_{C_1/E}\oplus ({\mathcal{N}}_{E/X_p}{)}_{|C_1}\simeq {\mathcal{O}}_{C_1}(1)\oplus {{\mathcal{O}}_E(-1)}_{|C_1},$$

$$K_{X_1}+D+E+E_1=g_1^{\ast }(K_{X_p}+D+E)$$

$$\Gamma\sim\sigma+4f\mbox{ where $f$ is a fibre of }\mathbb{F}_{2}\to\mathbb{P}^{1},\mbox{ and }\Gamma^{2}=6,\Gamma\cdot E_{|E_{1}}=2.$$

$$\{u=s_2=s_4=0\}.$$

$${\mathcal{N}}_{C_2/X_1}\simeq {\mathcal{N}}_{C_2/E}\oplus ({\mathcal{N}}_{E/X_2}{)}_{|C_2}={\mathcal{O}}_{C_2}(1)\oplus {\mathcal{O}}_{C_2}(-2).$$

$$E_2\simeq {\mathbb{P}}_{{\mathbb{P}}^1}({\mathcal{N}}_{C_2/X_1})\simeq {\mathbb{F}}_3.$$

$$K_{X_2}+D+E+E_1+E_2=g_2^{\ast }(K_{X_1}+D+E+E_1).$$

$$g_2\circ g_1\circ f:(\tilde{X},D_{\tilde{X}})=(X_2,D+E+E_1+E_2)\to (X,D_X)$$

$$X=\{x_3(x_0^3+x_1^3)+x_2^4+x_0{x}_1{x}_2{x}_3+x_4(x_3^3+x_4^3)=0\},$$

$$L\cap \{x_0^3+x_1^3=0\},$$

$$\{0\}\in \{x^3+y^3+z^4+xyz=0\}.$$

$$D_X⇢{\mathbb{P}}_{x_0,x_1,x_2}^2;$$

$$X=\{x_0^3{x}_3+x_1^4+x_2^4+x_0{x}_1{x}_2{x}_3+x_4(x_3^3+x_4^3)=0\}\subset {\mathbb{P}}^4$$

$$\{0\}\in \{x^3+y^4+z^4+xyz=0\}.$$

$$\{x_1^4+x_2^4=x_0(x_1{x}_2+x_0^2)=0\}.$$

Full text

1. Introduction and motivation

A Calabi–Yau (CY) pair $(X,D_{X})$ consists of a normal projective variety $X$ and a reduced sum of integral Weil divisors $D_{X}$ such that $K_{X}+D_{X}\sim_{\mathbb{Z}}0$.

The class of CY pairs arises naturally in a number of problems and comprises examples with very different birational geometry. Indeed, on the one hand, a Gorenstein Calabi–Yau variety $X$ can be identified with the CY pair $(X,0)$. On the other hand, if $X$ is a Fano variety, and if $D_{X}$ is an effective reduced anticanonical divisor, then $(X,D_{X})$ is also a CY pair.

Definition 1.1**.**

1. (a)

A pair $(X,D_{X})$ is (t,dlt) (resp. (t,lc)) if $X$ is $\mathbb{Q}$-factorial, terminal and $(X,D_{X})$ divisorially log terminal (resp. log canonical). 2. (b)

A birational map $(X,D_{X})\stackrel{{\scriptstyle\varphi}}{{\dashrightarrow}}(Y,D_{Y})$ is volume preserving if $a_{E}(K_{X}+D_{X})=a_{E}(K_{Y}+D_{Y})$ for every geometric valuation $E$ with centre on $X$ and on $Y$.

The dual complex of a dlt pair $(Z,D_{Z}=\sum D_{i})$ is the regular cell complex obtained by attaching an $(|I|-1)$-dimensional cell for every irreducible component of a non-empty intersection $\bigcap_{i\in I}D_{i}$.

The dual complex encodes the combinatorics of the lc centres of a dlt pair and [4] show that its PL homeomorphism class is a volume preserving birational invariant.

By [3, Theorem 1.9], a (t,lc) CY pair $(X,D_{X})$ has a volume preserving (t,dlt) modification $(\widetilde{X},D_{\widetilde{X}})\to(X,D_{X})$, and the birational map between two such modifications is volume preserving.

Abusing notation, I call dual complex the following volume preserving birational invariant of a (t,lc) CY pair $(X,D_{X})$.

Definition 1.2**.**

$\mathcal{D}(X,D_{X})$ is the PL homeomorphism class of the dual complex of a volume preserving (t,dlt) modification of $(X,D_{X})$.

As the underlying varieties of CY pairs range from CY to Fano varieties, they can have very different birational properties. However, $X$ being Fano is not a volume preserving birational invariant of the pair $(X,D_{X})$. Following [13], I consider the following volume preserving birational invariant notion:

Definition 1.3**.**

A (t,lc) CY pair $(X,D_{X})$ has maximal intersection if $\dim\mathcal{D}(X,D_{X})=\dim X-1$.

In other words, $(X,D_{X})$ has maximal intersection if there is a volume preserving (t,dlt) modification of $(X,D_{X})$ with a [math]-dimensional log canonical centre. Maximal intersection CY pairs have some Fano-type properties; Kollár and Xu show the following:

Theorem 1.4**.**

Let $(X,D_{X})$ be a dlt maximal intersection CY pair, then:

[13, Proposition 19]** $X$ is rationally connected, 2. 2.

[13, Theorem 21]** there is a volume preserving map $(X,D_{X})\stackrel{{\scriptstyle\varphi}}{{\dashrightarrow}}(Z,D_{Z})$ such that $D_{Z}$ fully supports a big and semiample divisor.

Remark 1.5**.**

The expression “Fano-type” should be understood with a pinch of salt. Having maximal intersection is a degenerate condition: a general (t,lc) CY pair $(X,D_{X})$ with $X$ Fano and $D_{X}$ a reduced anticanonical section needs not have maximal intersection.

Definition 1.6**.**

A toric pair $(X,D_{X})$ is a (t,lc) CY pair formed by a toric variety and the reduced sum of toric invariant divisors.

A toric model is a volume preserving birational map to a toric pair.

Example 1.7**.**

A CY pair with a toric model has maximal intersection.

Remark 1.8**.**

In dimension $2$, the converse holds: maximal intersection CY surface pairs are precisely those with a toric model [6].

The characterisation of CY pairs with a toric model is an open and difficult problem. A characterisation of toric pairs was conjectured by Shokurov and is proved in [1], but it is not clear how to refine it to get information on the existence of a toric model. A motivation to better understand the birational geometry of CY pairs and their relation to toric pairs comes from mirror symmetry.

The mirror conjecture extends from a duality between Calabi-Yau varieties to a correspondence between Fano varieties and Landau-Ginzburg models, i.e. non-compact Kähler manifolds endowed with a superpotential. Most known constructions of mirror partners rely on toric features such as the existence of a toric model or of a toric degeneration. In an exciting development, Gross, Hacking and Keel conjecture the following construction for mirrors of maximal intersection CY pairs.

Conjecture 1.9**.**

[6]** Let $(Y,D_{Y})$ be a simple normal crossings maximal intersection CY pair. Assume that $D_{Y}$ supports an ample divisor, let $R$ be the ring $k[\mathrm{Pic}(Y)^{\times}]$, $\Omega$ the canonical volume form on $U$ and

[TABLE]

Then, the free $R$-module $V$ with basis $U^{\mathrm{trop}}(\mathbb{Z})$ has a natural finitely generated $R$-algebra structure whose structure constants are non-negative integers determined by counts of rational curves on $U$.

Denote by $K$ the torus $\mathrm{Ker}\{\mathrm{Pic}(Y)\to\mathrm{Pic}(U)\}$. The fibration

[TABLE]

is a $T_{K}$-equivariant flat family of affine maximal intersection log CY varieties. The quotient

[TABLE]

only depends on $U$ and is the mirror family of $U$.

Versions of Conjecture 1.9 are proved for cluster varieties in [7], but relatively few examples are known.

The goal of this note is to present examples of maximal intersection CY pairs that do not admit a toric model and for which one can hope to construct the mirror partner proposed in Conjecture 1.9 (see Section 2 for a precise statement).

2. Auxiliary results on $3$-fold CY pairs

The examples in Section 3 are $3$-fold maximal intersection CY pairs whose underlying varieties are birationally rigid. In particular, such pairs admit no toric model; this shows that [6]’s results on maximal intersection surface CY pairs do not extend to higher dimensions. In this section, I first recall some results on birational rigidity of Fano $3$-folds. Then, I introduce the (t,dlt) modifications suited to the construction outlined in Conjecture 1.9 and discuss the singularities of the boundary $D_{X}$.

2.1. Birational rigidity

Let $X$ be a terminal $\mathbb{Q}$-factorial Fano $3$-fold. When $X$ has Picard rank $1$, $X$ is a Mori fibre space, i.e. an end product of the classical MMP.

Definition 2.1**.**

A birational map $Y/S\stackrel{{\scriptstyle\varphi}}{{\dashrightarrow}}Y^{\prime}/S^{\prime}$ between Mori fibre spaces $Y/S$ and $Y^{\prime}/S^{\prime}$ is square if it fits into a commutative square

[TABLE]

where $g$ is birational and the restriction $Y_{\eta}\stackrel{{\scriptstyle\varphi_{\eta}}}{{\dashrightarrow}}Y^{\prime}_{\eta}$ is biregular, where $\eta$ is the function field of the base $k(S)$.

A Mori fibre space $Y/S$ is (birationally) rigid if for every birational map $Y/S\stackrel{{\scriptstyle\varphi}}{{\dashrightarrow}}Y^{\prime}/S^{\prime}$ to another Mori fibre space, there is a birational self map $Y/S\stackrel{{\scriptstyle\alpha}}{{\dashrightarrow}}X/S$ such that $\varphi\circ\alpha$ is square.

In particular, if $X$ is a rigid Mori fibre space, then $X$ is non-rational and no (t,lc) CY pair $(X,D_{X})$ admits a toric model.

Non-singular quartic hypersurfaces $X_{4}\subset\mathbb{P}^{4}$ are probably the most famous examples of birationally rigid $3$-folds [9]. Some mildly singular quartic hypersurfaces are also known to be birationally rigid, in particular, we have:

Proposition 2.2**.**

[2, 16]** Let $X_{4}\subset\mathbb{P}^{4}$ be a quartic hypersurface with no worse than ordinary double points. If $|\operatorname{Sing}(X)|\leq 8$, then $X$ is $\mathbb{Q}$-factorial (in particular, $X$ is a Mori fibre space) and is birationally rigid.

2.2. Singularities of the boundary

I now state some results on the singularities of the boundary of a $3$-fold (t,lc) CY pair. Let $(X,D_{X})$ be a $3$-fold (t,lc) CY pair and $(\widetilde{X},D_{\widetilde{X}})$ a (t,dlt) modification. A stratum of $(\widetilde{X},D_{\widetilde{X}})$ is an irreducible component of a non-empty intersection of components of $D_{\widetilde{X}}$. Given a stratum $W$, there is a divisor $\operatorname{Diff}_{W}D_{\widetilde{X}}$ on $W$ such that $(W,\operatorname{Diff}_{W}D_{\widetilde{X}})$ is a lc CY pair and

[TABLE]

When $K_{\widetilde{X}}+D_{\widetilde{X}}$ is Cartier and $D_{\widetilde{X}}$ reduced, $\operatorname{Diff}_{W}D_{\widetilde{X}}$ is the sum of the restrictions of the components of $D_{\widetilde{X}}$ that do not contain $W$.

In particular, for any irreducible component $S$ of $D_{\widetilde{X}}$, the link of $[S]$ in $\mathcal{D}(X,D_{X})$ is the dual complex $\mathcal{D}(S,\operatorname{Diff}_{S}D_{\widetilde{X}})$. Therefore, if $(X,D_{X})$ has maximal intersection, so does $(S,\operatorname{Diff}_{S}D_{\widetilde{X}})$. By the results of [6], $(S,\operatorname{Diff}_{S}D_{X})$ then has a toric model.

As $X$ has terminal singularities, $X$ is normal and Cohen-Macaulay. Any Cartier component $S$ of the boundary $D_{X}$ is Cohen-Macaulay and satisfies Serre’s condition $S_{2}$. By [12, Proposition 16.9], $(S,\operatorname{Diff}_{S}D_{X})$ is semi log canonical (slc). In particular, if $X$ is Gorenstein and $D_{X}$ irreducible, $D_{X}$ has slc singularities.

I am particularly interested in producing examples of (t,lc) CY pairs for which the mirror partners proposed in Conjecture 1.9 (see also [8]) can be constructed; this motivates the following definition:

Definition 2.3**.**

A (t,dlt) modification $(\widetilde{X},D_{\widetilde{X}})\to(X,D_{X})$ is called good if $(\widetilde{X},D_{\widetilde{X}})$ is log smooth in the sense of log geometry, that is if the components of $D_{\widetilde{X}}$ are non-singular and if $\widetilde{X}$ has only cyclic quotient singularities.

An immediate consequence of the definition is that if $(\widetilde{X},D_{\widetilde{X}})\stackrel{{\scriptstyle f}}{{\to}}(X,D_{X})$ is a good (t,dlt) modification and $D_{X}=\sum_{i}D_{i}$, then

[TABLE]

where $E$ is reduced and $f$-exceptional, and the restriction of $f$ to $f_{*}^{-1}D_{i}$ is a resolution for all $i$.

Normal singularities Let $p\in\operatorname{Sing}(D_{i})$ be an isolated singularity lying on a single component of the boundary. The restriction $f_{i}\colon\widetilde{D}_{i}\to D_{i}$ is a resolution and we have:

[TABLE]

where $E$ is defined by $K_{\widetilde{X}}+f_{*}^{-1}D_{\widetilde{X}}+E=f^{*}(K_{X}+D_{X})$.

We now assume that $D_{i}$ is Cartier, as is the case when $X$ is Gorenstein and $D_{X}$ irreducible. Without loss of generality, assume that $\operatorname{Sing}(D_{i})=p$. Then, $p$ is canonical if $E\cap\widetilde{D}_{i}=\emptyset$, and elliptic otherwise. Indeed, let

[TABLE]

be the factorisation through the minimal resolution of $(p\in D_{i})$. Then, $q$ is either an isomorphism or an isomorphism at the generic point of each component of $E_{|\widetilde{D}_{i}}$ because $f$ is volume preserving. We have: $K_{\overline{D}_{i}}=\mu^{*}K_{D_{i}}-Z$, where the effective cycle $Z=q_{*}(E_{D_{i}})$ is either empty (and $p$ is canonical) or a reduced sum of $\mu$-exceptional curves (and $p$ is elliptic). In the second case, $Z\sim-K_{\widetilde{D}_{i}}$ is the fundamental cycle of $(p\in D_{i})$. If $Z$ is irreducible, it is reduced and has genus $1$; if not, every irreducible component of $Z$ is a smooth rational curve of self-intersection $-2$.

When $p$ is elliptic, $Z$ is reduced and $p$ is a Kodaira singularity [10, Theorem 2.9], i.e. a resolution is obtained by blowing up points of the singular fibre in a degeneration of elliptic curves; further, in Arnold’s terminology, the singularity $p$ is uni or bimodal.

Further, $p\in D_{i}$ is a hypersurface singularity (resp. a codimension $2$ complete intersection, resp. not a complete intersection) when $-3\leq Z^{2}\leq-1$ (resp. $Z^{2}=-4$, resp. $Z^{2}\leq-5$) [14]. When $-1\leq Z^{2}\leq-4$, normal forms are known for $p\in D_{i}$: Table 1 lists normal forms of slc hypersurface singularities, while normal forms of codimension $2$ complete intersections elliptic singularities are given in [19].

3. Examples of rigid maximal intersection $3$-fold CY pairs

All the examples below are $(t,lc)$ CY pairs $(X,D_{X})$ which admit no toric model. Except for Example 3.4, all underlying varieties $X$ are birationally rigid quartic hypersurfaces by Proposition 2.2; the underlying variety in Example 3.4 is a smooth cubic $3$-fold, and therefore non-rational.

3.1. Examples with normal boundary

Example 3.1**.**

Consider the CY pair $(X,D_{X})$ where $X$ is the nonsingular quartic hypersurface

[TABLE]

and $D_{X}$ is its hyperplane section $X\cap\{x_{4}=0\}$.

The quartic surface $D_{X}$ has a unique singular point $p=(1{:}0{:}0{:}0{:}0)$, and using the notation of Table 1, $p$ is locally analytically equivalent to a $T_{4,4,4}$ cusp

[TABLE]

$D_{X}$ is easily seen to be rational: the projection from the triple point $p$ is

[TABLE]

this map is the blowup of the $12$ points $\{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=x_{1}x_{2}x_{3}=0\}$, of which $4$ lie on each coordinate line $L_{i}=\{x_{i}=0\}$, for $i=1,2,3$.

I treat this example in detail and construct explicitly a good (t,dlt) modification of the pair $(X,D_{X})$.

Let $f\colon X_{p}\to X$ be the blowup of $p$, then $X_{p}$ is non-singular, the exceptional divisor $E$ satisfies $(E,\mathcal{O}_{E}(E))=(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(-1))$, and if $D$ denotes the proper transform of $D_{X}$, we have:

[TABLE]

Explicitly, the blowup $\mathcal{F}\to\mathbb{P}^{4}$ of $\mathbb{P}^{4}$ at $p$ is the rank $2$ toric variety $\mathrm{TV}(I,A)$, where $I=(u,x_{0})\cap(x_{1},\dots,x_{4})$ is the irrelevant ideal of $\mathbb{C}[u,x_{0},\dots,x_{4}]$ and $A$ is the action of $\mathbb{C}^{\ast}\times\mathbb{C}^{\ast}$ with weights:

[TABLE]

The equation of $X_{p}$ is

[TABLE]

while $E=\{u=0\}$ and $D=\{u(s_{1}^{4}+s_{2}^{4}+s_{3}^{4})+x_{0}s_{1}s_{2}s_{3}=0\}$. By construction, $E$ is the projective plane with coordinates $s_{1},s_{2},s_{3}$. Note that $(X_{p},D+E)$ is not dlt because $D\cap E=\{x_{0}s_{1}s_{2}s_{3}=0\}$ consists of $3$ concurrent lines $C_{1},C_{2},C_{3}$.

Consider $g_{1}\colon X_{1}\to X_{p}$ the blowup of the nonsingular curve

[TABLE]

The exceptional divisor of $g_{1}$ is a surface $E_{1}\simeq\mathbb{P}(\mathcal{N}_{C_{1}/X_{p}})$, and since $C_{1}\simeq\mathbb{P}^{1}$, the restriction sequence of normal bundles gives

[TABLE]

so that $E_{1}=\mathbb{F}_{2}$. Further,

[TABLE]

where, abusing notation, I denote by $D$ and $E$ the proper transforms of the divisors $D$ and $E$. The “restricted pair” on $E_{1}$ is a surface CY pair $(E_{1},(D+E)_{|E_{1}})$ by adjunction. By construction, $E\cap E_{1}$ is the negative section $\sigma$. The curve $\Gamma=D\cap E_{1}$ is irreducible, and since $(D+E)_{|E_{1}}$ is anticanonical, we have

[TABLE]

The divisors $D,E,E_{1}$ meet in two points, the dual complex $\mathcal{D}(X_{1},D+E+E_{1})$ is not simplicial it is a sphere $S^{2}$ whose triangulation is given by $3$ vertices on an equator. While not strictly necessary, we consider a further blowup to obtain a (t,dlt) pair with simplicial dual complex.

Denote by $C_{2}$ the proper transform of the curve

[TABLE]

Then $C_{2}\subset E\cap D$ is rational, and as above

[TABLE]

Let $g_{2}\colon X_{2}\to X_{1}$ be the blowup of $C_{2}$, then the exceptional divisor of $g_{2}$ is a Hirzebruch surface

[TABLE]

Still denoting by $D,E,E_{1}$ the strict transforms of $D,E,E_{1}$, we have:

[TABLE]

The pair $(X_{2},D+E+E_{1}+E_{2})$ is dlt; the composition

[TABLE]

is a good (t,dlt) modification.

The “restrictions” of $(\widetilde{X},D_{\widetilde{X}})$ to the component of the boundary are the following surface anticanonical pairs:

On $D$: $(E+E_{1}+E_{2})_{|D}$ is a cycle of $(-3)$-curves, the morphism $D\to D_{X}$ is the familiar resolution of the $T_{4,4,4}$ cusp singularity; 2. -

On $E$: $(D+E_{1}+E_{2})_{E}$ is the triangle of coordinate lines with self-intersections $(1,1,1)$; 3. -

On $E_{1}$: $(D+E+E_{2})_{E_{1}}$ is an anticanonical cycle with self-intersections $(5,-3,-1)$; 4. -

On $E_{2}$: $(D+E+E_{1})_{E_{2}}$ is an anticanonical cycle with self-intersections $(5,-3,0)$ (as above, $E_{|E_{2}}\sim\sigma$ is a negative section, ${E_{1}}_{|E_{2}}\sim f$ a fibre of $\mathbb{F}_{3}\to\mathbb{P}^{1}$, and $D_{|E_{2}}\sim 4f+\sigma$).

It follows that the dual complex $\mathcal{D}(X,D_{X})$ is PL homeomorphic to a tetrahedron and $(X,D_{X})$ has maximal intersection. Note that $(0\in D_{X})$ is a maximal intersection lc point, and since $D_{X}$ is a rational surface, it has a toric model.

Example 3.2**.**

Let $X$ be the hypersurface

[TABLE]

and $D_{X}$ its hyperplane section $X\cap\{x_{4}=0\}$.

The quartic $X$ has $3$ ordinary double points at the intersection points

[TABLE]

where $L$ is the line $\{x_{2}=x_{3}=x_{4}=0\}$. The singular locus of $D_{X}$ is $\operatorname{Sing}(X)\cup\{p\}$, where $p=(0{:}0{:}0{:}1{:}0)$ is a $T_{3,3,4}$ cusp, i.e. locally analytically equivalent to

[TABLE]

The quartic surface $D_{X}$ is rational; the projection of $D_{X}$ from $p$ is

[TABLE]

this map is defined outside of the $12$ points (counted with multiplicity) defined by $\{x_{2}^{4}=x_{0}^{3}+x_{1}^{3}+x_{0}x_{1}x_{2}=0\}$.

If $\widetilde{X}\stackrel{{\scriptstyle f}}{{\to}}X$ is the composition of the blowups at the ordinary double points and at $p$, $\widetilde{X}$ is smooth and $D_{\widetilde{X}}$ is non-singular, so that $f$ is a good (t,dlt) modification.

The minimal resolution of $p\in D_{X}$ is a rational curve with self intersection $C^{2}=-3$. Explicitly, taking the blowup of $X$ at $p$, the proper transform is a rational surface $D$. The exceptional curve is the preimage of a nodal cubic in $\mathbb{P}^{2}$ blown up at 12 points counted with multiplicities. Note that $(\widetilde{X},D+E)$ is not dlt, but in order to obtain a (t,dlt) modification, we just need to blowup the node of $D\cap E$ which is a nonsingular point of $\widetilde{X},D$ and $E$. The (t,dlt) modification of $(X,D_{X})$ in a neighbourhood of $p$ is good and the associated dual complex is $2$-dimensional.

The pair $(X,D_{X})$ has maximal intersection; but as in the previous examples, $X$ is rigid, so that $(X,D_{X})$ can have no toric model.

Example 3.3**.**

Let $X$ be the nonsingular quartic hypersurface

[TABLE]

and $D_{X}$ its hyperplane section $X\cap\{x_{4}=0\}$.

The surface $D_{X}$ has a unique singular point $p=(0{:}0{:}0{:}1{:}0)$ of $D_{X}$, which is a cusp $T_{3,4,4}$, i.e. is locally analytically equivalent to

[TABLE]

As in Example 3.1, $X$ is non-singular, and finding a good (t,dlt) modification of $(X,D_{X})$ will amount to taking a minimal resolution of the singular point of $D_{X}$. Let $X_{p}\to X$ be the blowup of $X$ at $p$; $X_{p}$ is non-singular and if $D$ denotes the proper transform of $D_{X}$, and $E$ the exceptional divisor, $D\cap E$ consists of $2$ rational curves of self intersection $-3$ and $-4$. These curves are the proper transforms of $\{x_{0}=0\}$ and of $\{x_{0}^{2}+x_{1}x_{2}\}$ under the blow up of $\mathbb{P}^{2}_{x_{0},x_{1},x_{2}}$ at the points

[TABLE]

The dual complex consists of $3$ vertices that are joined by edges and span $2$ distinct faces: $\mathcal{D}(X,D_{X})$ is PL homeomorphic to a sphere $S^{2}$ whose triangulation is given by $3$ vertices on an equator. The CY pair $(X,D_{X})$ has maximal intersection but no toric model.

3.2. Examples with non-normal boundary

Example 3.4**.**

This example is due to R. Svaldi. Consider the cubic $3$-fold

[TABLE]

where $q,q^{\prime}$ are homogeneous polynomials of degree $2$ in $x_{0},\cdots,x_{4}$. If the quadrics $q$ and $q^{\prime}$ are general and if

[TABLE]

then $X$ and $S=\{x_{3}=0\}\cap X$ and $T=\{x_{4}=0\}\cap X$ are nonsingular.

Let $D_{X}$ be the anticanonical divisor $S+T$. The curve $C=S\cap T=\Pi\cap X$ for $\Pi=\{x_{3}=x_{4}=0\}$ is a nodal cubic. It follows that both $(S,C)$ and $(T,C)$ are log canonical, and therefore so is $(X,D_{X})$.

Since $S$ and $T$ are smooth, $\operatorname{Sing}(D_{X})=S\cap T=C$, and if $p$ is the node of $C$, we have:

[TABLE]

Thus, $p\in D_{X}$ is a double pinch point, i.e. $p$ is locally analytically equivalent to $\{0\}\in\{x^{2}y^{2}-z^{2}=0\}$.

We now construct a good (t,dlt) modification of $(X,D_{X})$. Let $f\colon X_{C}\to X$ be the blowup of $X$ along $C$; $\operatorname{Sing}(X_{C})$ is an ordinary double point.

Indeed, let $\Pi=\{x_{3}=x_{4}=0\}$, then $f$ is the restriction to $X$ of the blowup $\mathcal{F}\to\mathbb{P}^{4}$, where $\mathcal{F}$ is the rank $2$ toric variety $\mathrm{TV}(I,A)$, where $I=(u,x_{0},x_{1},x_{2})\cap(x_{3},x_{4})$ is the irrelevant ideal of $\mathbb{C}[u,x_{0},\dots,x_{4}]$ and $A$ is the action of $\mathbb{C}^{\ast}\times\mathbb{C}^{\ast}$ with weights:

[TABLE]

The equation of $X_{C}$ is

[TABLE]

so that $X_{C}$ has a unique singular point at

[TABLE]

and this is a $3$-fold ordinary double point. In addition, denoting by $E_{f}=\{u=0\}\cap X_{C}$ the exceptional divisor, we have

[TABLE]

so that the pair $(X_{C},\widetilde{S}+\widetilde{T}+E_{f})$ is a (t,lc) CY pair.

The pair $(X_{C},\widetilde{S}+\widetilde{T}+E_{f})$ is not dlt as the boundary has multiplicity $3$ along the fibre $F$ over the node of $S\cap T$. The blowup of $F$ is not $\mathbb{Q}$-factorial, therefore in order to obtain a good (t,dlt) modification, we consider the divisorial contraction $g\colon\widetilde{X}\to X_{C}$ centred along $F$. This is obtained by (a) blowing up the node, (b) then blowing up the proper transform of $F$, (c) flopping a pair of lines with normal bundle $(-1,-1)$ and (d) contracting the proper transform of the $\mathbb{P}^{1}\times\mathbb{P}^{1}$ above the node to a point $\frac{1}{2}(1,1,1)$. The exceptional divisor of $g$ is denoted by $E_{g}$.

The pair $(\widetilde{X},\widetilde{S}+\widetilde{T}+\widetilde{E}_{f}+E_{g})$ is the desired (t,dlt) modification of $(X,D_{X})$, and it has maximal intersection. The dual complex is PL homeomorphic to a tetrahedron.

Example 3.5**.**

Let $X$ be the quartic hypersurface

[TABLE]

where $l$ (resp. $q$) is a general linear (resp. quadratic) form in $x_{0},\cdots,x_{3}$, and $f_{3}$ a general homogeneous form of degree $3$ in $x_{0},\cdots,x_{4}$. Let $D_{X}$ be the hyperplane section $X\cap\{x_{4}=0\}.$

As $l,q$ and $f_{3}$ are general, $X$ has $6$ ordinary double points. Indeed, denote by $L=\{x_{1}=x_{3}=x=4=0\}$ and $L^{\prime}=\{x_{2}=x_{3}=x=4=0\}$, then

[TABLE]

which consists of $3$ points on each of the lines. In the neighbourhood of each point $q_{i}$ (resp. $q^{\prime}_{i}$) for $i=1,2,3$, the equation of $X$ is of the form

[TABLE]

(and $D_{X}=\{t=0\}$) so that all singular points of $X$ are ordinary double points. The quartic hypersurface $X$ is birationally rigid by Proposition 2.2.

The surface $D_{X}$ is non-normal as it has multiplicity $2$ along $L$ and $L^{\prime}$. The point $p=L\cap L^{\prime}$ is locally analytically equivalent to

[TABLE]

so that $p\in D_{X}$ is a double pinch point. We conclude that the surface $D_{X}$ has slc singularities, and hence $(X,D_{X})$ is a (t,lc) CY pair.

We construct a good (t,dlt) modification as follows.

First, since $\operatorname{Sing}(X)\cap L$ (resp. $\operatorname{Sing}(X)\cap L^{\prime}$) is non-empty, the blowup of $X$ along $L$ (resp. along $L^{\prime}$) is not $\mathbb{Q}$-factorial. In order to remain in the (t,dlt) category, we consider the divisorial extraction $f\colon X_{L}\to X$ centered on $L$ (resp. $L^{\prime}$).This is obtained by (a) blowing up the $3$ nodes lying on $L$, (b) blowing up the proper transform of $L$, (c) flopping $3$ pairs of lines with normal bundle $(-1,-1)$ and (d) contracting the proper transforms of the three exceptional divisors $\mathbb{P}^{1}\times\mathbb{P}^{1}$ lying above the nodes to points $\frac{1}{2}(1,1,1)$. The exceptional divisor of $f$ is denoted by $E$. Let $p\colon\widetilde{X}\to X$ denote the morphism obtained by composing the divisorial extraction centered on $L$ with that centered on $L^{\prime}$ (in any order), and let $E,E^{\prime}$ denote the exceptional divisors of the divisorial extractions. Then

[TABLE]

is a (t,dlt) modification of $(X,D_{X})$ and it has maximal intersection. The dual complex $\mathcal{D}(X,D_{X})$ is is PL homeomorphic to a sphere $S^{2}$ whose triangulation is given by $3$ vertices on an equator.

4. Further results on quartic $3$-fold CY pairs: beyond maximal intersection

This section concentrates on (t,lc) CY pairs $(X,D_{X})$, where $X$ is a factorial quartic hypersurface in $\mathbb{P}^{4}$ and $D$ is an irreducible hyperplane section of $X$. I give some more detail on the possible dual complexes of such pairs.

As explained in Section 2.2, $D_{X}$ is slc because $(X,D_{X})$ is lc. In order to study completely the dual complexes of such (t,lc) CY pairs, one needs a good understanding of the normal forms of slc singularities that can lie on $D$. In the case of a general Fano $X$, this step would require additional work, but here, $D_{X}$ is a quartic surface in $\mathbb{P}^{3}$ and the study of singularities of such surfaces has a rich history. I recall some results directly relevant to the construction of degenerate $CY$ pairs $(X,D_{X})$. The classification of singular quartic surfaces in $\mathbb{P}^{3}$ can be broken in three independent cases.

1. (a)

Quartic surfaces with no worse than rational double points: the minimal resolution is a $K3$ surface. Possible configurations of canonical singularities were studied by several authors using the moduli theory of $K3$ surfaces; there are several thousands possible configurations. The pair $(X,D_{X})$ is (t,dlt) and the dual complex of $(X,D_{X})$ is reduced to a point. 2. (b)

Non-normal quartic surfaces were classified by Urabe [17]; there are a handful of cases recalled in Theorem 4.1. 3. (c)

Non-canonical quartic surfaces with isolated singularities. These are studied by Wall [20] and Degtyarev [5] among others; their results are recalled in Theorem 4.5.

Theorem 4.1**.**

[17*]*A non-normal quartic surface $D\subset\mathbb{P}^{3}$ is one of:

the cone over an irreducible plane quartic curve with a singular point of type $A_{1}$ or $A_{2}$. 2. 2.

a ruled surface over a smooth elliptic curve $G$, $D=\varphi_{\mathcal{L}}(Z)$, where: 3. (a)

$\mathcal{L}=\mathcal{O}_{Z}(C_{1})\otimes\pi^{*}M$, and $Z=\mathbb{P}_{G}(\mathcal{O}_{G}\oplus N)$, for

-$M$* a line bundle of degree $2$ and*

-$N$* a non-trivial line bundle of degree [math].*

Denoting by $L_{i}$ the images by $\varphi_{\mathcal{L}}$ of the sections of $Z$ associated to $\mathcal{O}_{G}\oplus N\to\mathcal{O}_{G}$ and $\mathcal{O}_{G}\oplus N\to N$, $\operatorname{Sing}(D)=L_{1}\cup L_{2}$. 4. (b)

$\mathcal{L}=\mathcal{O}_{Z}(C)\otimes\pi^{*}M$, and $Z=\mathbb{P}_{G}(E)$ for a rank $2$ vector bundle $E$ that fits in a non-splitting

[TABLE]

*Denoting by $L$ the image by $\varphi_{\mathcal{L}}$ of a section $G\to Z$, $\operatorname{Sing}D=L$. * 5. 3.

a rational surface $D\subset\mathbb{P}^{3}$ which is 6. (a)

the image of a smooth $S\subset\mathbb{P}^{5}$ under the projection from a line disjoint from $S$; $D$ has no isolated singular point and

-$S=v_{2}(\mathbb{P}^{2})$, where $v_{2}$ is the Veronese embedding; $D$ is the Steiner Roman surface and is homeomorphic to $\mathbb{R}\mathbb{P}^{2}$;

-$S=\varphi(\mathbb{P}^{1}\times\mathbb{P}^{1})$, where $\varphi$ is the embedding defined by $|l_{1}+2l_{2}|$ for $l_{1,2}$ the rulings of $\mathbb{P}^{1}\times\mathbb{P}^{1}$;

-$S=\varphi(\mathbb{F}_{2})$, where $\varphi$ is the embedding defined by $|\sigma+f|$ for $\sigma$ the negative section and $f$ the fibre of $\mathbb{F}_{2}$. 7. (b)

the image of a surface $\hat{D}\subset\mathbb{P}^{4}$ with canonical singularities under the projection from a point not lying on it; $\hat{D}$ is a degenerate dP$4$* surface which is the blowup of $\mathbb{P}^{2}$ in $5$ points in almost general position.* 8. (c)

a rational surface embedded by a complete linear system on its normalisation $\hat{D}$; the non-normal locus of $D$ is a line $L$ and $D$ may have isolated singularities outside $L$. The minimal resolution of the normalisation of $D$ is a blowup of $\mathbb{P}^{2}$ in $9$ points. The normalisation of $D$ has at most two rational triple points lying on the inverse image of the non-normal locus; their images on $D$ are also triple points.

Remark 4.2**.**

$D$ is not slc in case 1.

Corollary 4.3**.**

Let $(X,D_{X})$ be a (t,lc) quartic CY pair with non-normal boundary. Then, $(X,D_{X})$ has maximal intersection except in the cases described in 2.(a) and (b) of Theorem 4.1.

Example 4.4**.**

Consider the pair $(X,D_{X})$ where:

[TABLE]

where $q$ is a general quadratic form in $(x_{1},x_{2})$ and $f_{3}$ a general cubic in $x_{0},\cdots,x_{4}$.

When $q$ and $f_{3}$ are general, the quartic hypersurface $X$ has $3$ ordinary double points. Indeed, denote by $L=\{x_{0}=x_{1}=x_{4}=0\}$, then $\operatorname{Sing}(X)$ consists of points of intersection of $L$ with $\{f_{3}=0\}$; there are $3$ such points $\{q_{1},q_{2},q_{3}\}$ when $f_{3}$ is general. In the neighbourhood of each point $q_{i}$ for $i=1,2,3$, the equation of $X$ is of the form

[TABLE]

(and $D_{X}=\{t=0\}$) so that all singular points of $X$ are ordinary double points. The nodal quartic $X$ is terminal and $\mathbb{Q}$-factorial because it has less than $9$ ordinary double points; $X$ is birationally rigid by [2, 16].

Taking the divisorial extraction of the line $L$ is enough to produce a dlt modification $(\widetilde{X},D_{\widetilde{X}}+E)$ of $(X,D_{X})$; this shows that $(X,D_{X})$ does not have maximal intersection. The dual complex has a single 1-stratum, the elliptic curve $D_{\widetilde{X}}\cap E$, which is a $(2,2)$ curve in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. The quartic surface $D_{X}$ is a ruled surface over an elliptic curve isomorphic to $D_{\widetilde{X}}\cap E$; it is an example of case 2.(b) in Theorem 4.1.

Theorem 4.5**.**

[20]** A normal quartic surface $D\subset\mathbb{P}^{3}$ with at least one non-canonical singular point is one of:

$D$* has a single elliptic singularity and $D$ is rational, or* 2. 2.

$D$* is a cone, or* 3. 3.

$D$* is elliptically ruled and* 4. (a)

$D$* has a double point $p$ with tangent cone $z^{2}$, the projection away from $p$ is the double cover of $\mathbb{P}^{2}$ branched over a sextic curve $\Gamma$. The curve $\Gamma$ is the union of $3$ conics in a pencil that also contains a double line. When this line is a common chord, $D$ has two $T_{2,3,6}$ singularities, when this line is a common tangent, $D$ has one singularity of type $E_{4,0}$. In the first case, $D$ may have an additional $A_{1}$ singular point.* 5. (b)

$D$* is $\{(x_{0}x_{3}+q(x_{1},x_{2}))^{2}+f_{4}(x_{1},x_{2},x_{3})=0\}$ and $\{f_{4}=0\}$ is four concurrent lines. Depending on whether $L=\{x_{3}=0\}$ is one of these lines or not and on whether the point of concurrence lies on $L$, $D$ has either two $T_{2,4,4}$ singular points or one trimodal elliptic singularity. The surface may have additional canonical points $A_{n}$ for $n=1,2,3$ or $2A_{1}$.*

Example 4.6**.**

Let $X$ be the nonsingular quartic hypersurface

[TABLE]

and $D_{X}$ its hyperplane section $X\cap\{x_{4}=0\}$. The surface $D_{X}$ is normal,

[TABLE]

and each singular point is simple elliptic $J_{2,0}=T_{2,3,6}$, i.e. is locally analytically equivalent to $\{0\}\in\{x^{2}+y^{3}+z^{6}+xyz=0\}$.

Here $X$ is nonsingular and $D_{X}$ is irreducible and normal, and as I explain below, finding a good (t,dlt) modification amounts to constructing a minimal resolution of $D_{X}$. Let $\widetilde{X}\to X$ be the composition of the weighted blowups at $p=(1{:}0{:}0{:}0{:}0)$ with weights $(0,2,1,3,1)$ and at $p^{\prime}=(0{:}0{:}0{:}1{:}0)$ with weights $(3,1,2,0,1)$, and denote by $E$ and $E^{\prime}$ the corresponding exceptional divisors. Note that $\widetilde{X}$ is terminal and $\mathbb{Q}$-factorial by [11, Theorem 3.5] and has no worse than cyclic quotient singularities. The morphism

[TABLE]

is volume preserving and the intersection of $D$ with each exceptional divisor is a smooth elliptic curve $C_{6}\subset\mathbb{P}(1,1,2,3)$ not passing through the singular points of $E$ and $E^{\prime}$; $f$ is a good (t,dlt) modification.

The dual complex $\mathcal{D}(X,D_{X})$ is $1$-dimensional, it has $3$ vertices and $2$ edges; $(X,D_{X})$ does not have maximal intersection. The quartic surface $D_{X}$ is an example of case 3.(a) in Theorem 4.5.

Corollary 4.7**.**

Let $(X,D_{X})$ be a (t,lc) quartic CY pair. Assume that $D_{X}$ is normal, has non-canonical singularities but is not a cone. Then $(X,D_{X})$ has maximal intersection except in cases 3.(a) and (b) of Theorem 4.5.

Remark 4.8**.**

When $\dim\mathcal{D}(X,D_{X})=1$, $D_{X}$ either has two $T_{2,3,6}$ or two $T_{2,4,4}$ singularities. Indeed, as is explained in Section 2.2, singular points $p\in D$ are Kodaira singularities, and in particular are at worst bimodal. The description of cases 3.(a) and (b) of Theorem 4.5 immediately implies the result, because a surface singularity of type $E_{4,0}$ is trimodal.