Hyperelliptic Integrals and Mirrors of the Johnson-Koll\'ar del Pezzo Surfaces

TL;DR

This paper constructs explicit Landau-Ginzburg mirror models for a family of del Pezzo surfaces with empty anticanonical systems, using hypergeometric functions as periods of specific curve pencils.

Contribution

It introduces a novel mirror construction for del Pezzo surfaces with empty anticanonical systems via hypergeometric periods, expanding mirror symmetry methods.

Findings

The I-function is a period of an explicit pencil of curves.

The constructed pencil serves as a candidate LG mirror.

Connections to hypergeometric function theory are established.

Abstract

For all k>0 integer, we consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1), first constructed by Johnson and Koll\'ar. We show that this function, which is of hypergeometric type, is a period of an explicit pencil of curves. Thus the pencil is a candidate LG mirror of the family of del Pezzo surfaces. The main feature of these surfaces, which makes the mirror construction especially interesting, is that the anticanonical system is empty: because of this, our mirrors are not covered by any other construction known to us. We discuss connections to the work of Beukers, Cohen and Mellit on hypergeometric functions.

Equations

$$\widehat{I}_{k}(\alpha)=\sum_{j=0}^{\infty}\frac{\bigl{(}(8k+4)j\bigr{)}!j!}{(2j)!\bigl{(}(2k+1)j\bigr{)}!^{2}\bigl{(}(4k+1)j\bigr{)}!}\ \alpha^{j}$$

$$X_{8k+4}\subset \mathbb{P}(2,2k+1,2k+1,4k+1)$$

$$Y_k=\left(\alpha y^2-h_{k,\alpha }(t_0,t_1)=0\right)\subset \mathbb{P}(1,1,3k+2)\times {\mathbb{C}}^{\times }$$

$$h_{k,\alpha}(t_{0},t_{1})=t_{1}(4t_{1}^{2k+1}+\alpha t_{0}^{2k+1})\bigl{(}-64t_{1}^{4k+2}+t_{0}t_{1}^{4k+1}-32\alpha t_{0}^{2k+1}t_{1}^{2k+1}-4\alpha^{2}t_{0}^{4k+2}\bigr{)}$$

$${\alpha }_{k,0}=\frac{{(4k+1)}^{4k+1}}{4^{8k+3}{(2k+1)}^{2(2k+1)}}$$

$${\pi }_k(\alpha )=\frac{1}{2\pi \text{i}}{\oint }_{{\gamma }_{k,\alpha }}\frac{t^{2k}dt}{y}$$

$$W_k=\left(\alpha \cdot (u_1+u_2+u_3+u_4-1)-{u_1}^2{u_2}^{2k+1}{u_3}^{2k+1}{u_4}^{4k+1}=0\right)\subset {\mathbb{T}}^4\times {\mathbb{C}}^{\times }$$

$${\mathrm{gr}}_3^W{R}^3{\upsilon }_{U_k!}\mathbb{Q}(1)=R^1{w}_{U_k\star }\mathbb{Q}$$

$$K_X={\mathcal{O}}_X(d-\sum _{i=0}^m{a}_i).$$

$$\widehat{I}_{X}(\alpha)=\sum_{j=0}^{\infty}\frac{(d\,j)!\,j!}{(a_{0}j)!\dots(a_{m}j)!}\,\alpha^{j}\quad\big{(}\alpha\in\mathbb{C}\big{)}$$

$$P_0(j)=-\prod _{i=0}^m(a_{i}j)(a_{i}j-1)\cdots (a_{i}j-a_i+1)\text{and}{P}_1(j-1)=j(dj)(dj-1)\cdots (dj-d+1)$$

$$\pi (\alpha )={\int }_{{\gamma }_{\alpha }}{\omega }_{\alpha }(\alpha \in D^{\times })$$

$$\hat{I}(\alpha )=\pi (\alpha )$$

$$\mathcal{E}=(R^{n-1}{w}_{U\star }\mathbb{Q})\otimes {\mathcal{O}}_U.$$

$$R^{n-1}{w}_{U\star }\mathbb{C}={\underline{\mathrm{Hom}}}_{{\mathcal{D}}_U}({\mathcal{O}}_U,\mathcal{E}),\text{and}{R}^{n-1}{w}_{U!}\mathbb{C}={\underline{\mathrm{Hom}}}_{{\mathcal{D}}_U}(\mathcal{E},{\mathcal{O}}_U)$$

$${\mathcal{D}}_U\cdot \omega \subset \mathcal{E},\text{and}\gamma :{\mathcal{D}}_U\cdot \omega \to {\mathcal{D}}_U\cdot \pi$$

$${\pi }_k(\alpha )=\frac{1}{2\pi \text{i}}{\oint }_{{\gamma }_{k,\alpha }}{\omega }_{k,\alpha }$$

$$h_{k,\alpha }(t)=t\left(4t^{2k+1}+\alpha \right)\left(-64t^{4k+2}+t^{4k+1}-32\alpha t^{2k+1}-4{\alpha }^2\right)$$

$$\underset{\alpha \to 0}{\lim }{g}_{k,\alpha }(t)=t^{4k+1}(-64t+1)$$

$$t_{\alpha }^{4k+1}=4(4t_{\alpha }^{2k+1}+\alpha {)}^2$$

$$r^{\frac{2}{4k+1}}\ll {\rho }_k(r)\ll r^{\frac{1}{2k+1}}$$

$${\pi }_k(\alpha )=\sum _{j=0}^{\infty }\left(\genfrac{}{}{0ex}{}{-1/2}{m}\right)(-1{)}^m{4}^{3m-j}\sum _{p=j}^{2m}(\genfrac{}{}{0ex}{}{-1/2}{p-j}\left)\right(\genfrac{}{}{0ex}{}{2m}{p}){\alpha }^j$$

$${\pi }_k(\alpha )=\frac{1}{2\pi \text{i}}{\oint }_{{\overline{\gamma }}_{k,\alpha }}\frac{1}{\sqrt{\left(1+\frac{4t^{2k+1}}{\alpha }\right)}}\frac{1}{\sqrt{\left(1-t{\left(8+\frac{2\alpha }{t^{2k+1}}\right)}^2\right)}}\frac{dt}{t}.$$

$$\begin{array}{rl}{\left(1+\frac{4t^{2k+1}}{\alpha }\right)}^{-\frac{1}{2}}& =\sum _{n=0}^{\infty }\left(\genfrac{}{}{0ex}{}{-1/2}{n}\right)\frac{4^n{t}^{(2k+1)n}}{{\alpha }^n}\\ {\left(1-t{\left(8+\frac{2\alpha }{t^{2k+1}}\right)}^2\right)}^{-\frac{1}{2}}& =\sum _{m=0}^{\infty }\left(\genfrac{}{}{0ex}{}{-1/2}{m}\right)(-1{)}^m{4}^{3m}{t}^m{\left(1+\frac{\alpha }{4t^{2k+1}}\right)}^{2m}\end{array}$$

$$\begin{array}{r}{\pi }_k(\alpha )=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{p=0}^{2m}\left(\genfrac{}{}{0ex}{}{-1/2}{n}\right)\left(\genfrac{}{}{0ex}{}{-1/2}{m}\right)\left(\genfrac{}{}{0ex}{}{2m}{p}\right)(-1{)}^m{4}^{3m+n-p}{\alpha }^{p-n}\frac{1}{2\pi \text{i}}{\oint }_{{\overline{\gamma }}_{k,\alpha }}\frac{dt}{t}{t}^{m-(2k+1)(p-n)}.\end{array}$$

$$\binom{-1/2}{m}(-1)^{m}4^{3m-j}\sum_{p=j}^{2m}\binom{-1/2}{p-j}\binom{2m}{p}=\frac{\bigl{(}(8k+4)j\bigr{)}!j!}{(2j)!\bigl{(}(2k+1)j\bigr{)}!^{2}\bigl{(}(4k+1)j\bigr{)}!}\ \,$$

$$(2l-1)!!=\frac{(2l)!}{2^l\cdot l!}\left(l>0\text{integer}\right)$$

$$\left(\genfrac{}{}{0ex}{}{-1/2}{m}\right)=\frac{(-1{)}^m(2m-1)!!}{2^{m}m!}=\frac{(-1{)}^m}{4^m}\left(\genfrac{}{}{0ex}{}{2m}{m}\right)$$

$$\sum _{p=j}^{2m}\left(\genfrac{}{}{0ex}{}{-1/2}{p-j}\right)\left(\genfrac{}{}{0ex}{}{2m}{p}\right)=\sum _{i=0}^{2m-j}\left(\genfrac{}{}{0ex}{}{-1/2}{i}\right)\left(\genfrac{}{}{0ex}{}{2m}{j+i}\right)=\sum _{i=0}^{2m-j}\left(\genfrac{}{}{0ex}{}{-1/2}{i}\right)\left(\genfrac{}{}{0ex}{}{2m}{2m-j-i}\right)=\left(\genfrac{}{}{0ex}{}{2m-1/2}{2m-j}\right)$$

Full text

[TABLE]

is a period of a pencil of curves of genus $3k+1$. The function $\widehat{I}_{k}$ is the regularised $I$-function of the family of anticanonical del Pezzo hypersurfaces $X=X_{8k+4}\subset\mathbb{P}(2,2k+1,2k+1,4k+1)$ and the pencil we construct is a candidate LG mirror of the elements of the family. The surfaces $X$ were first constructed by Johnson and Kollár [17]. The main feature of these surfaces, which makes the mirror construction especially interesting, is that $|-K_{X}|=|\mathcal{O}_{X}(1)|=\emptyset$; thus, there is no way to form a Calabi–Yau pair $(X,D)$ out of $X$. We also discuss the connection between our constructions and the work of Beukers, Cohen and Mellit [1] on hypergeometric functions.

1. 1 Introduction
2. 2 Anticanonical weighted hypersurfaces and mirror symmetry
3. 3 Proof of the main Theorem
4. 4 Relations to the work of Beukers, Cohen, Mellit
5. 5 Proof of Theorem 6
6. A Toric MMP

1 Introduction

We begin by stating our results; then we briefly comment on the context and on the methods of our proofs; we conclude with a few open questions.

1.1 Results

We state our results; more detail will be given in Section 2 and in Section 4.

For all $k>0$ integer, consider the hypergeometric function defined by the power series:

[TABLE]

The function $\widehat{I}_{k}$ satisfies a hypergeometric differential equation of order $6k+2$, see Remark 13.

Conjecturally $\widehat{I}_{k}$ is (a shift of) Givental’s $\widehat{G}$-function — a generating function of certain Gromov–Witten invariants, see Remark 8 — of the family of anticanonical del Pezzo hypersurfaces

[TABLE]

These surfaces form the main series of the classification of anticanonical quasi-smooth and well-formed two-dimensional weighted hypersurfaces [17], see Section 2.2.

In this paper we give positive answers to the following questions which, as explained in Section 2.1, for us are equivalent:

Question 1**.**

Is $\widehat{I}_{k}$ a period of a pencil of curves?

Question 2**.**

Does the family of anticanonical del Pezzo hypersurfaces of Equation (2) have a Landau–Ginzburg (LG) mirror?

For all $k>0$ integer, our answer to both questions is given by the one-parameter family of hyperelliptic curves:

[TABLE]

where $t_{0},t_{1},y$ are coordinates on $\mathbb{P}(1,1,3k+2)$, $\alpha$ is a coordinate on $\mathbb{C}^{\times}$ and

[TABLE]

together with the projection $w_{k}\colon Y_{k}\to\mathbb{C}^{\times}$ to the second factor.

Remark 3**.**

Let

[TABLE]

For $\alpha\neq\alpha_{k,0}$, the fibre $Y_{k,\,\alpha}=w_{k}^{-1}(\alpha)$ is a nonsingular hyperelliptic curve of genus $3k+1$.

Our first result is:

Theorem 4** (Main Theorem).**

For all $k>0$ integer, $\widehat{I}_{k}(\alpha)=\pi_{k}(\alpha)$, where

1. (A)

$\widehat{I}_{k}(\alpha)$* is the hypergeometric function of Equation (1) above, and* 2. (B)

$\pi_{k}(\alpha)$* is the period of the family $Y_{k}$ (3) given by:*

[TABLE]

where $\gamma_{k,\alpha}\subset Y_{k,\,\alpha}$ is the explicit cycle described in Section 3 and $t=t_{1}/t_{0}$.

The solutions of the differential equation satisfied by $\widehat{I}_{k}$ are sections of an irreducible complex local system on $U_{k}=\mathbb{P}^{1}\setminus\{0,\alpha_{k,0},\infty\}$, which we denote by $\mathbb{H}_{k}^{\text{red}}$, see Remark 13.

Write $Y_{U_{k}}=w_{k}^{-1}(U_{k})$ and denote by $w_{U_{k}}\colon Y_{U_{k}}\to U_{k}$ the restriction.

By the main Theorem and Remark 12 $\mathbb{H}_{k}^{\text{red}}$ is a subquotient of the local system $R^{1}w_{U_{k}\,\star}\mathbb{C}$. Since the two local systems have the same rank, we obtain:

Corollary 5**.**

For all $k>0$ integer, $\mathbb{H}_{k}^{\text{red}}$ is isomorphic to $R^{1}w_{U_{k}\,\star}\mathbb{C}$.

Our next result connects our construction to the work of Beukers, Cohen and Mellit [1]. More detail on this discussion can be found in Section 4. For all $k>0$ integer, consider the manifold:

[TABLE]

where $\mathbb{T}^{4}\cong(\mathbb{C}^{\times})^{4}$ is a $4$-dimensional torus with coordinates $u_{1},\dots,u_{4}$ and $\alpha$ is a coordinate on $\mathbb{C}^{\times}$. Denote by $\upsilon_{k}\colon W_{k}\to\mathbb{C}^{\times}$ the second projection. The 3-dimensional hypersurface $W_{k,\alpha}=\upsilon_{k}^{-1}(\alpha)\subset\mathbb{T}^{4}$ is nonsingular if and only if $\alpha\neq\alpha_{k,0}$.

A very special case of the work [1] relates point counting in characteristic $p$ on $W_{k,\alpha}$ to finite analogs of the function $\widehat{I}_{k}$ of Equation (1). This result strongly suggests that the hypergeometric function $\widehat{I}_{k}$ is a period of the family of $3$-folds $W_{k,\alpha}$ ($\alpha\in U_{k}$) or, equivalently, that the variation $H^{3}(W_{k,\alpha},\mathbb{Q})$ is related to the local system $\mathbb{H}_{k}^{\text{red}}$.333It would not be very difficult — but it would take us too far — to prove that $\widehat{I}_{k}$ is a period of the family of Equation (7). In Section 4 we prove:

Theorem 6**.**

For all $k>0$ integer, write $W_{U_{k}}=\upsilon_{k}^{-1}(U_{k})$ and denote by $\upsilon_{U_{k}}\colon W_{U_{k}}\to U_{k}$ the restriction. Then:

[TABLE]

This gives an interpretation of a special case of the results in [1] in terms of mirror symmetry.

1.2 Context and a few words on our proofs

Our mirrors of the surfaces $X=X_{8k+4}\subset\mathbb{P}(2,2k+1,2k+1,4k+1)$ are not covered by any mirror construction known to us. Indeed, since $H^{0}(X,-K_{X})=H^{0}(X,\mathcal{O}_{X}(1))=(0)$, there is no Calabi–Yau pair $(X,D)$, and hence the intrinsic mirror symmetry program [9] is not applicable in this context.

By [18, Theorem 5.4.4] the local systems $\mathbb{H}^{\text{red}}_{k}$ support a canonical rational variation of Hodge structures (VHS).444In fact, [18] constructs an explicit geometric realisation of this VHS, different from the one of Equation (7). By the criterion of [4, 8], the variation has Hodge weight one. Thus, it is natural to ask — even as there is no reason to expect it — if $\mathbb{H}^{\text{red}}_{k}$ is a (direct summand of) the variation of $H^{1}$ of a one-parameter family of curves: this motivates our Question 1. Since an irreducible local system supports at most one rational VHS, see [5, Proposition 2.1], our main Theorem implies that $\mathbb{H}^{\text{red}}_{k}=R^{1}w_{U_{k}\,\star}\mathbb{Q}$ as VHS.

The general shape of the Fano/LG correspondence suggests that the mirror of the family of anticanonical del Pezzo hypersurfaces $X_{8k+4}$ is a function $w_{k}\colon Y_{k}\to\mathbb{C}^{\times}$ with one-dimensional fibres, smooth over $U_{k}$, together with an identification of a subquotient of $R^{1}w_{U_{k}\,!}\mathbb{C}$ with the hypergeometric local system of solutions of the differential equation satisfied by $\widehat{I}_{k}$. This motivates our Question 2.

Our proof of the main Theorem is elementary: we expand the period in power series with the help of the residue theorem. The key difficulty is to find the equation of $Y_{k}$ (and the integration cycles $\gamma_{k}$).

The work [1] and in particular the pencil of $3$-folds $\upsilon_{k}\colon W_{k}\to\mathbb{C}^{\times}$ of Equation (7) are the starting point for our investigations. Morally, this work identifies the local system $\mathbb{H}^{\text{red}}_{k}$ with $\mathrm{gr}_{3}^{W}R^{3}\upsilon_{U_{k}\,!}\mathbb{C}$ — more detail on this point can be found in Section 4 — and thus provides a mirror of the wrong dimension. Then it is natural to ask if there is a morphism $w_{k}\colon Y_{k}\to\mathbb{C}^{\times}$ with one-dimensional fibres such that $\mathrm{gr}_{3}^{W}R^{3}\upsilon_{U_{k}\,!}\mathbb{Q}\,(1)=R^{1}w_{U_{k}\,!}\mathbb{Q}$. Our Theorem 6 states that our family of hyperelliptic curves indeed has this property. The constructions in the proof of the Theorem in Section 5 make it clear how the curve $Y=Y_{k,\alpha}$ arises naturally from a study of the geometry of the $3$-fold $W=W_{k,\alpha}$ $\forall\alpha\neq\alpha_{k,0}$. The fact that the variation $\mathrm{gr}_{3}^{W}H^{3}(W_{k,\alpha},\mathbb{Q})$ is the variation of the $H^{1}$ of a pencil of curves is nontrivial. There is no reason to expect it. Looking for an explanation, one is lead to wonder whether the $3$-folds $W_{k,\alpha}$ are rational. In Section A.3 we construct a conic bundle in the Mori category, birational to $W_{k,\alpha}$. Interestingly, a study of the geometry of this conic bundle suggests that $W_{k,\alpha}$ is not rational, and that the conic bundle may even be birationally rigid.

The idea of the proof of Theorem 6 is to construct a partial compactification $W\subset\widehat{W}$ with a del Pezzo fibration $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$. The del Pezzo fibration becomes visible after a monomial substitition of coordinates, see Equation (26). In the Appendix, we explain how we discovered the del Pezzo fibration by running a minimal model program for a partial resolution of a compactification of $W$. Once we have the del Pezzo fibration, the proof of Theorem 6 is an exercise in mixed Hodge theory for which models exist in the literature. The key point is Lemma 24, which constructs an algebraic cycle $Z\subset\operatorname{CH}_{2}(Y\times_{\mathbb{C}^{\times}}\widehat{W})$ inducing an isomorphism $p_{\star}\mathbb{Q}_{Y}\to R^{2}\phi_{\star}\mathbb{Q}_{\widehat{W}}(1)$.

It is natural to ask if our computation of the cohomology of $W$ in terms of the del Pezzo fibration $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$ really is the easiest way to prove Theorem 6. We think that it is. The conic bundle birational to $W$ that we construct in Section A.3 gives another, more complicated, way to understand the geometry of $W$.

1.3 Further questions

It may be possible to determine all hypergeometric functions that are periods of a pencil of curves, and describe the pencils explicitly.

There are several contexts where a construction of mirror symmetry is available that produces a mirror of the wrong dimension: one of these is general toric complete intersections [11, 1]; another is the abelian/nonabelian correspondence [2]. Our ideas here can form the basis of a systematic method to extract from these constructions mirrors of the correct dimension. In particular it would be extremely attractive to eliminate the assumptions of Hori–Vafa [11] and obtain mirrors for all toric complete intersections.

It will be interesting to see that our mirrors satisfy Homological Mirror Symmetry.

1.4 Structure of the paper

All sections in this paper logically depend on the Introduction; other than that, they are logically mutually independent and can be read (or not read, as the case may be) in any order.

The paper is structured as follows. In Section 2 we expand on the notion of mirror symmetry for Fano anticanonical weighted hypersurfaces. The families of surfaces (2) are an example of such hypersurfaces; our main Theorem can be interpreted as giving mirrors of these families. In Section 3 we prove the main Theorem. In Section 4 we show how our Equation (7) arises from the work [1] as a mirror of the wrong dimension. In Section 5 we prove Theorem 6. In the Appendix we explain how we discovered the del Pezzo fibration $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$. The Appendix ends with the construction of a compactification $W\subset W^{\prime}$ with a conic bundle structure $\psi\colon W^{\prime}\to\mathbb{F}_{1}$.

1.5 Acknowledgments

It is a pleasure to thank Fernando Rodriguez Villegas for attracting our attention to the work of Beukers, Cohen and Mellit [1], and Tom Coates and Don Zagier for helpful conversations. A.C. is partially supported by EPSRC Program Grant Classification, Computation and Construction: New Methods in Geometry; G.G. is supported by EPSRC-funded Centre for Doctoral Traning The London School of Geometry and Number Theory.

2 Anticanonical weighted hypersurfaces and mirror symmetry

In this Section we expand on the notion of mirror symmetry for Fano anticanonical weighted hypersurfaces. The families of surfaces $X=X_{8k+4}$ of the Introduction are an example of such hypersurfaces; our main Theorem can be interpreted as giving mirrors of these families.

We denote by $\mathbb{P}(a_{0},a_{1},\dots,a_{m})$ the weighted projective $m$-space with weights $a_{0}\leq a_{1}\leq\cdots\leq a_{m}$; we simply write $\mathbb{P}$ when $m$ and the weights are clear from the context.

For $X=X_{d}\in|\mathcal{O}_{\mathbb{P}}(d)|$ a quasismooth [12, 6.3] and wellformed [12, 6.10] hypersurface of degree $d$, the adjuction formula for the canonical sheaf $K_{X}$ [12, 6.14] states that

[TABLE]

By definition $X$ is Fano if and only if $-K_{X}$ is ample, i.e., if and only if $d<\sum_{i=0}^{m}a_{i}$. We say that $X$ is anticanonical if $-K_{X}=\mathcal{O}_{X}(1)$, that is, $d=\sum_{i=0}^{m}a_{i}-1$.

2.1 Our notion of mirror symmetry

We state what we mean by mirror for a quasismooth wellformed Fano anticanonical weighted hypersurface $X=X_{d}\subset\mathbb{P}(a_{0},\dots a_{m})$ of dimension $n=m-1$.

Definition 7**.**

The regularised $I$-function of $X$ is defined as the hypergeometric series

[TABLE]

Remark 8**.**

The paper [3, Sections B,C] defines the $G$-function of $X$, a generating series for certain Gromov–Witten invariants of $X$, and the $I$-function of $X$.555In [3] $X$ is a smooth variety. Here we think of a quasi-smooth well-formed weighted hypersurface as a smooth Deligne–Mumford stack. The definitions of $I_{X}$ and ${G}_{X}$ make sense in this more general context. The function $\widehat{I}_{X}$ (9) is the Fourier transform of the $I$-function of $X$. Conjecturally, ${G}_{X}(\alpha)=e^{-c\alpha}{I}_{X}(\alpha)$, where $c$ is the only rational number such that the right-hand side has the form $1+O({\alpha}^{2})$, see [3, Proposition D.9].

Remark 9**.**

Our functions $\widehat{I}_{X}$ satisfy a hypergeometric differential equation on $\mathbb{P}^{1}$, nonsingular outside $\Sigma=\{0,\alpha_{0},\infty\}$ where $\alpha_{0}=\frac{\prod a_{i}^{a_{i}}}{d^{d}}$, whose solutions are the sections of an irreducible complex local system on $\mathbb{P}^{1}\setminus\Sigma$ which we denote by $\mathbb{H}_{X}^{\text{red}}$. To be a little more specific, write

[TABLE]

Consider the differential operator $H_{X}=P_{0}(D)+\alpha P_{1}(D)\in\mathbb{Z}[\alpha,D]$ where $D=\alpha\frac{d}{d\alpha}$, and denote by $H^{\text{red}}_{X}$ the operator obtained removing from both $P_{0}(j)$ and $P_{1}(j-1)$ a copy of every common factor. By [18, Corollary 3.2.1] $H^{\text{red}}_{X}$ is irreducible and it is easy to see that $H^{\text{red}}_{X}\cdot\widehat{I}_{X}=0$.

Definition 10**.**

A Landau–Ginzburg (LG) model is a tuple $(Y^{n},w,\omega,\gamma)$ where:

1. (i)

$Y$ is a smooth algebraic manifold of dimension $n$; 2. (ii)

$w\colon Y\to\mathbb{C}^{\times}$ is a quasi-projective morphism. For $\alpha\in\mathbb{C}^{\times}$ we denote by $Y_{\alpha}=w^{-1}(\alpha)$ the fibre.

Denote by $U\subset\mathbb{C}^{\times}$ the set of regular values of $w$, and by $w_{U}\colon Y_{U}=w^{-1}(U)\to U$ the restriction.

1. (iii)

$\omega\in\Gamma(U,w_{U\,\star}\Omega^{n-1}_{Y_{U}/U})$. For $\alpha\in U$, we write $\omega_{\alpha}\in H^{0}(Y_{\alpha},\Omega^{n-1}_{Y_{\alpha}})$ the corresponding form; 2. (iv)

$\gamma\in\Gamma(D^{\times},R^{n-1}w_{U\,!}\mathbb{Q}_{Y_{U}})$ where $0\in D\subset\mathbb{C}$ is a small disk and $D^{\times}=D\setminus\{0\}$. For $\alpha\in D^{\times}$ we denote by $\gamma_{\alpha}\in H_{n-1}(Y_{\alpha},\mathbb{Q})$ the corresponding cycle.

The period of the LG model is the function

[TABLE]

Definition 11**.**

Let $X=X_{d}\subset\mathbb{P}(a_{0},\dots a_{m})$ be a quasismooth wellformed Fano anticanonical weighted hypersurface of dimension $n=m-1$. A LG model $(Y^{n},w,\omega,\gamma)$ is mirror of $X$ if for all $\alpha$

[TABLE]

Remark 12**.**

It follows directly from Definition 11 that $\mathbb{H}_{X}^{\text{red}}$ is a subquotient of $R^{n-1}w_{U\,!}\mathbb{C}$ and this fact could be taken as a weak version of mirror symmetry.

Indeed, write

[TABLE]

The sheaf $\mathcal{E}$ carries an algebraic connection $\nabla\colon\mathcal{E}\to\Omega^{1}_{U}\otimes\mathcal{E}$ (the Gauss–Manin connection) by means of which we can regard it as a $\mathcal{D}_{U}$-module, where $\mathcal{D}_{U}$ denotes the sheaf of differential operators on $U,$ and

[TABLE]

are the local systems of flat sections and of solutions of $\mathcal{E}$. Now $w_{U\,\ast}\Omega^{n-1}_{Y_{U}/U}\subset\mathcal{E}$ as the last piece of the Hodge filtration and we regard $\omega$ as a section of $\mathcal{E}$ by means of this inclusion. On the other hand we regard $\gamma$ as a solution of $\mathcal{E}$ and recover the period as $\pi=\gamma(\omega)$. We have an inclusion and a surjection of $\mathcal{D}_{U}$-modules:

[TABLE]

So we have $\mathbb{H}_{X}^{\text{red}}=\operatorname{\underline{Hom}}_{\mathcal{D}_{U}}(\mathcal{D}\cdot\pi,\mathcal{O}_{U})\subset\operatorname{\underline{Hom}}_{\mathcal{D}_{U}}(\mathcal{D}_{U}\cdot\omega,\mathcal{O}_{U})$ and $R^{n-1}w_{U\,!}\mathbb{C}=\operatorname{\underline{Hom}}_{\mathcal{D}_{U}}(\mathcal{E},\mathcal{O}_{U})\twoheadrightarrow\operatorname{\underline{Hom}}_{\mathcal{D}_{U}}(\mathcal{D}_{U}\cdot\omega,\mathcal{O}_{U})$.

2.2 Anticanonical del Pezzo hypersurfaces

In this paper we call a Fano surface a del Pezzo surface.

Johnson and Kollár [17] classify all anticanonical quasismooth wellformed del Pezzo surfaces in weighted projective $3$-spaces. Their classification consists of $22$ sporadic cases and the series (2), where $k\in\mathbb{N},\ k>0$. The 22 sporadic cases are all listed in [17, Theorem 8].

By (9) for all $k>0$ integer the regularised $I$-function of any surface of the family (2) is given by Equation (1).

Remark 13**.**

By Remark 9 the function $\widehat{I}_{k}$ (1) satisfies an hypergeometric differential equation on $\mathbb{P}^{1}$ which is singular on $\Sigma_{k}=\{0,\alpha_{k,0},\infty\}$, where $\alpha_{k,0}$ is as in Equation (5). The reduced differential operator associated to $\widehat{I}_{k}$, which we denote by $H_{k}^{\mathrm{red}},$ has order $6k+2,$ thus the local system $\mathbb{H}_{k}^{\mathrm{red}}$ given by its solutions has rank $6k+2.$

Remark 14**.**

In light of the definitions of Section 2.1, the data at the beginning of Section 3 define a LG model and our main Theorem can be interpreted as stating that this LG model is the mirror to the corresponding family of Johnson and Kollár. Also, by Remarks 12 and 13, $\mathbb{H}_{k}^{\mathrm{red}}\simeq R^{1}w_{U_{k}\,\star}\mathbb{C}$, as stated in Corollary 5.

3 Proof of the main Theorem

In this Section, we prove the main Theorem stated in the Introduction. We begin by giving data to construct the period integral.

The period integral

Fix an integer $k>0$. We define data $(Y_{k},w_{k},\omega_{k},\gamma_{k})$ as follows:

1. (i)

$Y_{k}\subset\mathbb{P}(1,1,3k+2)\times\mathbb{C}^{\times}$ is the $2$-dimensional manifold given by (3) and (4); 2. (ii)

$w_{k}\colon Y_{k}\to\mathbb{C}^{\times}$ is the projection on the second factor.

Let $\alpha_{k,0}$ be as in Equation (5) and consider $\alpha\neq\alpha_{k,0}$.

1. (iii)

$\omega_{k,\alpha}=\frac{1}{2\pi\text{{i}}}\frac{t^{2k}\,dt}{y}$, where $t=t_{1}/t_{0}$; 2. (iv)

$\gamma_{k,\alpha}$ is the cycle that we describe next.

From this data we construct the period integral:

[TABLE]

This is the period of $Y_{k}$ of Equation (6).

Remark 15**.**

This data defines a LG model, according to Definition 10, and the period (6) is the period of the LG model (10). Our main Theorem can be interpreted as stating that this LG model is a mirror of the family of surfaces $X_{k}$.

The cycle of integration

Let us denote by $p_{k,\alpha}\colon Y_{k,\alpha}\to\mathbb{P}^{1}$ the $2:1$ cover; $p_{k,\alpha}$ is branched at the $6k+4$ roots of the polynomial

[TABLE]

Lemma 16**.**

For $|\alpha|\ll 1$ the polynomial $h_{k,\alpha}$ has:

• •

A root at $t=0$;

• •

$4k+1$* roots of norm $\sim|\alpha|^{\frac{2}{4k+1}}$;*

• •

$2k+1$* roots of norm $\sim|\alpha|^{\frac{1}{2k+1}}$;*

• •

A root of norm $\sim\frac{1}{64}$.

Proof.

Clearly $h_{k,\alpha}$ has a root at $t=0$ and $2k+1$ roots of norm $|\alpha/4|^{\frac{2}{4k+1}}$. Consider the polynomial $g_{k,\alpha}(t)=-64t^{4k+2}+t^{4k+1}-32\alpha t^{2k+1}-4\alpha^{2}$; then as $\alpha\to 0$ $g_{k,\alpha}(t)$ has a root $t_{\alpha}\sim\frac{1}{64}$ and $4k+1$ roots $t_{\alpha}$ of norm $\sim|\alpha|^{\frac{2}{4k+1}}$. Indeed

[TABLE]

hence for $|\alpha|\ll 1$ $g_{k,\alpha}$ has a root that tends to $\frac{1}{64}$ and $4k+1$ roots that tend to [math]. Now $t_{\alpha}$ is a root if and only if

[TABLE]

and if $t_{\alpha}\to 0$, then $|t_{\alpha}|^{4k+1}\sim|\alpha|^{2}$. ∎

Choose a continuous function $\rho_{k}\colon(0,1)\to\mathbb{R}$ such that for $r\ll 1$:

[TABLE]

For $|\alpha|\ll 1$ let $\overline{\gamma}_{k,\alpha}$ be the circle of radius $\rho_{k}(|\alpha|)$ around the origin in $\mathbb{C}$ starting at $t_{0}=\rho_{k}(|\alpha|)$. By Lemma 16 this circle divides $\mathbb{P}^{1}$ into two regions each containing an even number of branch points of $p_{k,\alpha}$, see Figure 1. Hence $p_{k,\alpha}^{-1}(\overline{\gamma}_{k,\alpha})\subset Y_{k,\alpha}$ consists of two disjoint circles: we take $\gamma_{k,\alpha}$ to be the lift along which the power series expansion in Equation (11) is valid, see Figure 2. 666Alternatively we choose the base point for our lift such that $y>0$.

Proof of the main Theorem

An immediate consequence of the next two lemmas. ∎

Lemma 17**.**

Let $\pi_{k}(\alpha)$ be the period integral described above. Then, setting $m=(2k+1)j$:

[TABLE]

Proof.

By a small manipulation we write the period as:

[TABLE]

By the defining inequalities of the function $\rho_{k}$, and our choice of $\gamma_{k,\alpha}$, both the following power series expansions hold:

[TABLE]

Plugging the two power series in (12), switching the series and the integral signs and using the binomial theorem we obtain:

[TABLE]

Finally, applying the residue theorem around $t=0$ and setting $j=p-n,$ we obtain the result. ∎

Lemma 18**.**

For all $k>0$ and $j\geq 0$ integers, setting $m=(2k+1)j$:

[TABLE]

Proof.

Clearly (13) holds when $j=0,$ thus we assume $j\geq 1$. In what follows we repeatedly use the identity:

[TABLE]

We have:

[TABLE]

We set $i=p-j$ and we write the finite sum on the left hand side of (13) as:

[TABLE]

where the last equality in (16) follows from the Chu–Vandermonde formula for generalised binomial coefficients:

[TABLE]

Plugging (15) and (16) in (13) and using that $m=(2k+1)j$ we can rewrite (13) as:

[TABLE]

Now we note that

[TABLE]

Using this equality combined with (14) for $l=j$, we simplify (17) as:

[TABLE]

Since (18) manifestly holds (it is (14) for $l=(4k+2)j$), the result is proved. ∎

4 Relations to the work of Beukers, Cohen, Mellit

In this Section we show how our Equation (7) arises from the work [1] as a mirror of the wrong dimension.

4.1 Finite hypergeometric functions and point counting

We summarise the main result in [1].

Let $v,w\in\mathbb{Q}^{d}$ be such that $\forall i,j\in\{1,\dots,d\}$ $v_{i}\neq w_{j}\ \mathrm{mod}\ \mathbb{Z}$ and the polynomials $\prod_{j=1}^{d}(x-e^{2\pi iv_{j}})$ and $\prod_{j=1}^{d}(x-e^{2\pi iw_{j}})$ are products of cyclotomic polynomials. Then there exist natural numbers $p_{1},\dots,p_{r}$ and $q_{1},\dots,q_{s},$ with $p_{1}+\dots+p_{s}=q_{1}+\dots+q_{s},$ such that

[TABLE]

and the analytic hypergeometric function

[TABLE]

can be rewritten in the form:

[TABLE]

Similarly, the finite hypergeometric function $H_{q}(v,w|\lambda)$ — where $q$ is a prime power coprime with $\operatorname{hcf}(v,w)$ — can be written in terms of $p_{i},q_{j}$ only [1, Definition 1.1, Theorem 1.3].

For all $\alpha\in\mathbb{F}_{q}^{\times},$ Beukers, Cohen and Mellit consider the quasiprojective (in fact affine) variety $W_{\alpha}$ given by the homogeneous equations:

[TABLE]

and prove the following, see [1, Theorem 1.5] for the precise statement and [1, Section 5] for its proof:

Theorem 19**.**

If $\mathrm{gcd}(p_{1},\dots,p_{r},q_{1},\dots,q_{s})=1$ and $M\cdot\alpha\neq 1,$ there exists a nonsingular completion $\overline{W}_{\alpha}$ of $W_{\alpha}$ such that $|\overline{W}_{\alpha}(\mathbb{F}_{q})|=H_{q}(v,w,M\cdot\alpha)$ up to factors depending only on $p_{i},q_{j}.$

4.2 The family of $3$-folds of Theorem 6

We provide some additional context for our statement of Theorem 6. Specifically, we build a precise connection with Theorem 19.

Fix $k>0$ integer. We specialise the above discussion to the pair of vectors $v_{k},w_{k}$ in $\mathbb{Q}^{d_{k}},\ d_{k}=6k+2$:

[TABLE]

This leads to

[TABLE]

For $\lambda=M_{k}\cdot\alpha$, by (19)

[TABLE]

that is, the regularised $I$-function of Equation (1).

Still following the above discussion, for all $\alpha$ in $\mathbb{C}^{\times}$ consider the quasiprojective (in fact, affine) $3$-fold $W_{k,\,\alpha}$ given by the homogeneous equations:

[TABLE]

Next, we manipulate the Equations (20) to rewrite them as in the Introduction. By solving the first Equation for $y_{2}$ and writing $x_{0}=y_{1}$, for all $\alpha$ in $\mathbb{C}^{\times}$ the system of Equations (20) leads to the Equation of a $3$-fold in $\mathbb{P}^{4}$:

[TABLE]

The $3$-fold $W_{k,\,\alpha}$ is the intersection of the projective $3$-fold defined by the Equation above with the torus $\mathbb{T}^{4}\subset\mathbb{P}^{4}$ and, in the affine chart $\{x_{0}\neq 0\}$ with coordinates $u_{i}=x_{i}/x_{0}$, it is described by the Equation:

[TABLE]

This is the same as Equation (7); hence the meaning of the symbol $W_{k,\alpha}$ is the same as in the Introduction.

Theorem 19 computes $|\overline{W}_{k,\alpha}(\mathbb{F}_{q})|$ in terms of the finite analog of $\widehat{I}_{k}$. The statement can be interpreted as asserting that the manifold $W_{k}$ of Equation (7) is a sort of LG mirror of the family of del Pezzo surfaces $X_{8k+4}\subset\mathbb{P}(2,2k+1,2k+1,4k+1)$ of the wrong dimension. Indeed, by [6] the only interesting cohomology group of $\overline{W}_{k,\alpha}$ occurs in degree $3$, hence by the Weil conjectures the main contribution to $|\overline{W}_{k,\alpha}(\mathbb{F}_{q})|$ comes from $H^{3}(\overline{W}_{k,\alpha},\mathbb{Q})$. Presumably for this completion we also have $H^{3}(\overline{W}_{k,\alpha},\mathbb{Q})=\mathrm{gr}^{W}_{3}H^{3}(W_{k,\alpha},\mathbb{Q})$ — as we do in Lemma 21.

5 Proof of Theorem 6

In this Section we prove Theorem 6, stated in the Introduction.

Fix $k>0$ integer and let $U_{k}=\mathbb{C}^{\times}\setminus\{\alpha_{k,0}\}$, as in the Introduction. We prove that for all $\alpha\in U_{k}$ there is an isomorphism of pure Hodge structures:

[TABLE]

Notation

For convenience in what follows we set $a=1/\alpha$. We write Equation (7) as

[TABLE]

and Equation (3) as

[TABLE]

where $h_{k,\alpha}$ is as in Equation (4). We denote by $W_{k,a}$ and $Y_{k,a}$ the fibres of (22) and (23) over $a$, so $W_{k,a}=W_{k,\alpha}$ and $Y_{k,a}=Y_{k,\alpha}$ where $\alpha=1/a$.

For the rest of this Section we fix $k>0$ and $a\neq a_{k,0}$, where $a_{k,0}=1/\alpha_{k,0}$, and we we omit all reference to $k$ and $a$. Also, we write $p_{\mathbb{C}^{\times}}\colon Y_{\mathbb{C}^{\times}}=p^{-1}(\mathbb{C}^{\times})\to\mathbb{C}^{\times}$ for the restriction of the $2:1$ cover $p\colon Y\to\mathbb{P}^{1}$.

Proof of Theorem 6

Let $W\subset\widehat{W}$ be the partial compactification of Equation (28) and $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$ the projective degree 2 del Pezzo fibration of Lemma 22. Applying in sequence Lemmas 21, 23, 24 we see:

[TABLE]

and the latter group is $H^{1}(Y,\mathbb{Q})(-1)$. ∎

The partial compactification

We construct a partial compactification $\widehat{W}\subset\mathbb{P}(1,1,2,2)/_{\mu_{2}}\times\mathbb{C}^{\times}$ of $W$ (where $\mu_{2}$ acts on $\mathbb{P}(1,1,2,2)$ as described below). Lemma 21 states that $\mathrm{gr}_{3}^{W}H^{3}_{c}(W,\mathbb{Q})=\mathrm{gr}_{3}^{W}H^{3}_{c}(\widehat{W},\mathbb{Q})$ hence for our purpose we might as well work with $\widehat{W}$ in place of $W$. The advantage of working with $\widehat{W}$ is that, as stated in Lemma 22, the second projection $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$ is a fibration with fibres del Pezzo surfaces, and the point of the substitution (26) is precisely to make this structure manifest. (In Appendix A we explain how we discovered the del Pezzo fibration structure, and the substitution, by the methods of the minimal model program.)

Consider the weighted projective space $\mathbb{P}(1,1,2,2)$ with weighted homogeneous coordinates $x_{1},x_{2},y_{1},y_{2}$, and the quotient $\mathbb{P}(1,1,2,2)/\mu_{2}$ where the group $\mu_{2}$ acts on the affine coordinates $x=x_{2}/x_{1},y=y_{1}/x_{1}^{2},z=y_{2}/x_{1}^{2}$ by

[TABLE]

Here and in what follows we write

[TABLE]

and we denote by $t$ the coordinate on $\mathbb{C}^{\times}$. Note that $G$ is a (noncompact) toric variety and the 4-dimensional torus $\mathbb{T}_{G}\subset G$ is the locus $\left(x,y,z,t\neq 0\right)$ in the affine piece $x_{1}\neq 0$. The substitution

[TABLE]

identifies $W$ with

[TABLE]

We denote by $\widehat{W}$ the closure of $W$ in $G$, given by the weighted homogeneous equation:

[TABLE]

The $3$-fold $\widehat{W}$ is a partial compactification of $W$.

Remark 20**.**

The $3$-fold $\widehat{W}$ is quasismooth and we think at it as a smooth orbifold. More precisely $\widehat{W}$ has nonisolated quotient singularities: although it is singular, it is a rational homology manifold. Because of this, for the purpose of cohomological computations, we can pretend that $\widehat{W}$ is smooth. Below we take the convention that the set of regular values of the map $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$ is the set of values $t\in\mathbb{C}^{\times}$ such that the corresponding fibre $\widehat{W}_{t}$ is quasismooth.

Lemma 21**.**

There is an identity of pure Hodge structures:

[TABLE]

Proof.

Note first that the $3$-fold $W$ is nonsingular but noncompact, thus $H^{3}_{c}(W,\mathbb{Q})$ is a mixed Hodge structure with weights $\leq 3$, and so is $H^{3}_{c}(\widehat{W},\mathbb{Q})$. Consider the divisor $D=\widehat{W}\setminus W$, and denote by $i\colon D\hookrightarrow\widehat{W}$ and $j\colon W\hookrightarrow\widehat{W}$ the inclusions. We have a long exact sequence of mixed Hodge structures

[TABLE]

To prove (29), we check that $\mathrm{gr}_{3}^{W}H^{2}_{c}(D,\mathbb{Q})=\mathrm{gr}_{3}^{W}H^{3}_{c}(D,\mathbb{Q})=(0)$. To this end, we study the geometry of the surface $D$; $D$ is the union $D=\bigcup_{i=1}^{4}\ D_{i}$, where:

[TABLE]

One can check that $D\subset\widehat{W}$ is (locally the quotient of) a simple normal crossing divisor with no [math]-dimensional stata. By setting

[TABLE]

we get a strict simplicial resolution $D^{[2]}\rightrightarrows D^{[1]}\rightarrow D$ and the long exact sequence:

[TABLE]

It follows that $H^{2}_{c}(D,\mathbb{Q})$ has weights $\leq 2.$ Now choose $m=3$ in (30) and examine $H^{3}_{c}(D,\mathbb{Q})$. On the left hand side, $\bigoplus_{i<j}H^{2}_{c}(D_{ij},\mathbb{Q})$ has weights $\leq 2.$ On the right hand side, $\bigoplus_{i}H^{3}_{c}(D_{i},\mathbb{Q})$ a priori has weights $\leq 3$. To conclude, we next show that, in fact, for $i=1,2,3,4,$ $H^{3}_{c}(D_{i},\mathbb{Q})$ has weights $<3$. Consider first the surface $D_{1}$, given by

[TABLE]

This is the same as the surface

[TABLE]

where $z_{2},y_{1},y_{2}$ are homogeneous coordinates of $\mathbb{P}^{2}$ and $\mu_{2}$ acts as $(y_{1},y_{2})\mapsto(-y_{1},-y_{2})$ on the affine piece $\left(z_{2}=1\right)$. Note that the quotient of $\mathbb{P}^{2}$ by the $\mu_{2}$-action is the weighted projective space $\mathbb{P}(1,1,2)$ with homogeneous coordinates $y_{1},y_{2},w_{2}=z_{2}^{2}$. In $\mathbb{P}(1,1,2)$ (31) becomes

[TABLE]

thus, since $t\neq 0$, we conclude that $D_{1}\simeq\mathbb{P}^{1}\times\mathbb{C}^{\times}$ and then $H^{3}_{c}(D_{1},\mathbb{Q})$ is a pure Hodge structure of weight $2$. An almost identical argument holds for $D_{2}.$ The surface $D_{4}$ is given by

[TABLE]

which is the same as the surface

[TABLE]

where $z_{1},z_{2},w_{1}$ are homogeoneous coordinates on $\mathbb{P}^{2},$ thus also $D_{4}\simeq\mathbb{P}^{1}\times\mathbb{C}^{\times}$ and $H^{3}_{c}(D_{4},\mathbb{Q})$ is a pure Hodge structure of weight $2$. To conclude, consider the surface $D_{3}$, given by

[TABLE]

By means of the substitution $y_{2}\mapsto y_{2}-x_{1}x_{2}/2$ write this as

[TABLE]

and note that this is the same as the surface

[TABLE]

where $z_{1}=x_{1}^{2},z_{2}=x_{2}^{2},w_{2}=y_{2}^{2}$ are homogeneous coordinates on $\mathbb{P}(1,1,2)$. A natural compatification of (32) is the surface $\overline{D}_{3}$ given by

[TABLE]

Note that $\overline{D}_{3}$ is a $2:1$ cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ branched along a divisor in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(4,2)|$, thus $\overline{D}_{3}$ is a rational surface and hence $H^{3}(\overline{D}_{3},\mathbb{Q})=\left(0\right)$. Then $H^{3}({D}_{3},\mathbb{Q})$ has weights $\leq 2$, since, by setting $\Gamma=\overline{D}_{3}\setminus D_{3}$, we have the long exact sequence of mixed Hodge structures

[TABLE]

and $\mathrm{gr}_{3}^{W}H^{2}({\Gamma},\mathbb{Q})=\left(0\right)$. ∎

Lemma 22**.**

Let $\phi\colon\widehat{W}\rightarrow\mathbb{C}^{\times}$ be the projection onto the second factor. Denote by $\Delta$ the set of critical values of $\phi$ and write $\Omega=\mathbb{C}^{\times}\setminus\Delta$ for the set of regular values. Let $j\colon\Omega\hookrightarrow\mathbb{C}^{\times}$ be the natural inclusion and denote by $\phi_{\Omega}:\widehat{W}_{\Omega}=\phi^{-1}(\Omega)\to\Omega$ the induced morphism. Let $\delta_{1},\delta_{2},\delta\in\mathbb{C}[t]$ be the polynomials:

[TABLE]

Write $\Omega_{1}=\mathbb{C}^{\times}\setminus\{\delta_{1}=0\}$, $\widehat{W}_{\Omega_{1}}=\phi^{-1}(\Omega_{1})$. Denote by $K$ the function field $k(\mathbb{C}^{\times})=\mathbb{C}(t)$. Then:

1. (1)

$\Delta=\{\delta=0\}$. If $t$ is a root of $\delta_{1}$, $\widehat{W}_{t}$ has a unique non quasismooth point $p_{t}=(0:0:-2:1)$; if $t$ is a root of $\delta_{2}$, $\widehat{W}_{t}$ has a unique non quasismooth point

[TABLE]

In both cases the non quasismooth point is an ordinary double point. 2. (2)

After the change of coordinates:

[TABLE]

the equation of the fibre $\widehat{W}_{t}$ over $t\in\Omega_{1}$ is:

[TABLE] 3. (3)

For all $t\in\Omega$ the fibre $\widehat{W}_{t}$ is a quasismooth del Pezzo surface of degree $2$ with two $1/4\cdot(1,-1)$ points $p^{+}_{t},p^{-}_{t}$, on $(x_{1}=x_{2}=0)$, and intersecting $(y_{1}=y_{2}=0)$ in two points $q^{+}_{t}$ and $q^{-}_{t}$. In the coordinates of Equation (36),

[TABLE] 4. (4)

For all $t\in\Omega$ the fibre $\widehat{W}_{t}$ has Picard rank $r=h^{2}(\widehat{W}_{t},\mathbb{Q})=2$. More specifically, $\widehat{W}_{t}$ contains a configuration of lines as pictured in Figure 3, and a basis of $\operatorname{Pic}(\widehat{W}_{t})_{\mathbb{Q}}$ is given the classes of the curves $C_{t,1}$ and $C_{t,2}$. 5. (5)

In the variables of Equation (36), the restriction $\phi_{1}\colon Y_{1}=\widehat{W}\cap\left(x_{1}=x_{2}=0\right)\to\mathbb{C}^{\times}$ is a $2:1$ branched cover with branch locus $\{\delta_{1}=0\}$, and the restriction $\phi_{2}\colon Y_{2}=\widehat{W}_{\Omega_{1}}\cap\left(y_{1}=y_{2}=0\right)\to\Omega_{1}$ is a $2:1$ branched cover with branch locus $\{\delta_{2}=0\}$. In particular,

[TABLE] 6. (6)

The Picard rank of $\widehat{W}_{K}$ is $\rho=1$. 7. (7)

$R^{3}\phi_{\star}\mathbb{Q}=R^{1}\phi_{\star}\mathbb{Q}=(0)$. 8. (8)

The natural homomorphism $R^{2}\phi_{\star}\mathbb{Q}\to j_{\star}j^{\star}R^{2}\phi_{\star}\mathbb{Q}=j_{\star}R^{2}\phi_{\Omega\,\star}\mathbb{Q}$ is an isomorphism.

Proof.

To prove (1), fix $t\in\mathbb{C}^{\times}$ and compute the Jacobian of the polynomial in Equation (28). To prove that for all $t\in\Delta$ the non quasismooth point of $\widehat{W}_{t}$ is an ordinary double point, one can check that the Hessian of (28) at that point is invertible.

Assertion (2) is a simple substitution.

To prove (3), fix $t\in\Omega$ and consider the quotient map $\sigma\colon\mathbb{P}(1,1,2,2)\to\mathbb{P}(1,1,2,2)/\mu_{2}$. Let $\widehat{V}_{t}=\sigma^{-1}(\widehat{W}_{t})$ be the preimage. Note that:

[TABLE]

since $\sigma$ is $2:1$ and etále in codimension 1. We have $-K_{\widehat{V}_{t}}=\mathcal{O}(2)$ and $-K_{\widehat{V}_{t}}^{2}=4$, thus $\widehat{V}_{t}\subset\mathbb{P}(1,1,2,2)$ is a del Pezzo surface of degree $4$. The orbifold points of $\widehat{V}_{t}$, of type $1/2\cdot(1,1)$, are the two points $P_{t}^{\pm}$ of $\widehat{V}_{t}$ on the line $\left(x_{1}=x_{2}=0\right)$; in the coordinates of Equation (36),

[TABLE]

Note that, by (24), on the affine piece $(y_{1}=1)$ the group $\mu_{2}$ acts by

[TABLE]

Hence $\widehat{W}_{t}\subset\mathbb{P}(1,1,2,2)/\mu_{2}$ is a del Pezzo surface of degree $2$ with two $1/4\cdot(1,-1)$ points $p^{\pm}_{t}=\sigma(P_{t}^{\pm})$, as in (37); these are the only orbifold points of $\widehat{W}_{t}$ since $(1:0:0:0)$ and $(0:1:0:0)$ do not satisfy (36). Setting $y_{1}=y_{2}=0$ in Equation (36), one finds that $\widehat{W}_{t}\cap\left(y_{1}=y_{2}=0\right)$ is given by the two points $q_{t}^{\pm}$ in (37).

To prove (4), note that the crepant resolution $\widetilde{W}_{t}$ of $\widehat{W}_{t}$ is a smooth weak del Pezzo surface of degree $2$. By Demazure’s Theorem this is the blow-up of $\mathbb{P}^{2}$ in $7$ nongeneral points, thus it has Picard rank $r=8$. Then the surface $\widehat{W}_{t}$, obtained by blowing down $6$ exceptional curves on $\widetilde{W}_{t}$, has Picard rank $r=2$. We explain how the geometry of $\widehat{W}_{t}$ singles out a distinguished basis of generators of ${\mathrm{Pic}(W_{t})}_{\mathbb{Q}}$. Setting

[TABLE]

on $\mathbb{P}(1,1,2,2)$ Equation (36) factors as:

[TABLE]

This exhibits eight lines on $\widehat{V}_{t}$, four of which passing through $P_{t}^{+}$, the other four passing thorugh $P_{t}^{-}$, as pictured on the left of Figure 4. The four lines on $\widehat{V}_{t}$ through $P_{t}^{+}$ correspond to two orbits under the $\mu_{2}$ action, and the same holds for the four lines through $P_{t}^{-}$. Namely, in $\mathbb{P}(1,1,2,2)/\mu_{2}$ Equation (36) can only be factored as:

[TABLE]

This exhibits four lines on the surface $\widehat{W}_{t}$, as pictured in the middle of Figure 4, two of which passing through $p_{t}^{+}:$

[TABLE]

the other two passing through $p_{t}^{-}$:

[TABLE]

To see that $C_{t,1}\sim C_{t,4}$ note that, on $\widetilde{W}_{t}$, the union of the strict transform $\widetilde{C}_{t,1}$ and the three exceptional curves above $p_{t}^{+}$, as pictured on the right of Figure 4, supports a fibre of a conic bundle where another fibre is supported on the union of the strict transform $\widetilde{C}_{t,4}$ and the three exceptional curves above $p_{t}^{-}$. Taking direct image this implies indeed that $C_{t,1}\sim C_{t,4}$. Similarly, $C_{t,2}\sim C_{t,3}$. Then the two curves $C_{t,1}$ and $C_{t,2}$ form a basis of $\mathrm{Pic}(\widehat{W}_{t})_{\mathbb{Q}}.$

Statement (5) follows immediately from Equation (36), and by (1) and (3). In particular, to prove  (38) one can check by (35) that, in the variables of Equation (28), if $t$ is a root of $\delta_{1}$ then $\phi_{1}^{-1}(t)=(0:0:2:-1)=p_{t}$, and if t is a root of $\delta_{2}$ then $\phi_{2}^{-1}(t)$ is given by

[TABLE]

which is the point $q_{t}$ in (34).

We prove (6) as follows. Note that the function fields of $Y_{1}$ and $Y_{2}$ are $k(Y_{1})=K(\sqrt{\delta_{1}})$ and $k(Y_{2})=K(\sqrt{\delta_{2}})$. Set $L=K(\sqrt{\delta_{1}},\sqrt{\delta_{2}})$ and consider the lattice of Galois field extensions:

[TABLE]

The Galois group $\mathrm{Gal}(L/K)\simeq C_{2}\times C_{2}$ is generated by the two authomorphisms $\sigma_{1}$ and $\sigma_{2}$, where:

[TABLE]

By (4), $\mathrm{Pic}(\widehat{W}_{L})$ is generated by the classes of the curves $C_{1}$ and $C_{2}$, of Equation (39). Then by Galois descent $\mathrm{Pic}(\widehat{W}_{K})$ has only one generator, given by the class of the curve $C_{1}+C_{2}+C_{3}+C_{4}\subset\widehat{W}_{K}$, since $C_{1}+C_{2}+C_{3}+C_{4}$ is the only curve in $\widehat{W}_{L}$ which is invariant with respect to the induced action of $\mathrm{Gal}(L/K)$.

Assertion (7) follows from the fact that, by $(1)$ and $(4)$, $\forall\ t\in\mathbb{C}^{\times}$ $H^{1}(\widehat{W}_{t},\mathbb{Q})=H^{3}(\widehat{W}_{t},\mathbb{Q})=(0)$.

To prove (8), consider a singular value $s\in\Delta$ and the fibre $\widehat{W}_{s}$. To check that $R^{2}\phi_{\star}\mathbb{Q}\to j_{\star}R^{2}\phi_{\Omega\,\star}\mathbb{Q}$ is an isomorphism in a neighbourhood of $s$, it is enough to show that the natural homomorphism

[TABLE]

is an isomorphism, where $t\in\mathbb{C}^{\times}$ is near $s$, $T_{s}\colon H^{2}(\widehat{W}_{t},\mathbb{Q})\to H^{2}(\widehat{W}_{t},\mathbb{Q})$ is the local monodromy operator at $s$, and $H^{2}(\widehat{W}_{t},\mathbb{Q})^{T_{s}}$ denotes the group of monodromy invariants. Indeed on the one hand, by the proper base change theorem, $H^{2}(\widehat{W}_{s},\mathbb{Q})$ is the fibre at $s$ of $R^{2}\phi_{\star}\mathbb{Q}$, and on the other hand $H^{2}(\widehat{W}_{t},\mathbb{Q})^{T_{s}}$ is the fibre at $s$ of $j_{\star}R^{2}\phi_{\Omega\,\star}\mathbb{Q}$.

We have an exact triangle of constructible complexes on $\widehat{W}_{s}$:

[TABLE]

where $\psi$ and $\varphi$ are the nearby and vanishing cycle functors [10, Exposé I]. Since $\widehat{W}_{s}$ has isolated hyperquotient singularities — in fact by (4) it only has one non quasismooth point — $\varphi\,\mathbb{Q}$ is supported at the non quasismooth point of $\widehat{W}_{s}$ and is concentrated in degree $2$ [19]. Thus from the exact sequence:

[TABLE]

we conclude that the natural homomorphism $H^{2}(\widehat{W}_{s},\mathbb{Q})\to H^{2}(\widehat{W}_{t},\mathbb{Q})$ is injective. By the local invariant cycle Theorem [7, Theorem 1.4.1] then $H^{2}(\widehat{W}_{s},\mathbb{Q})$ is the group of monodromy invariant cycles in $H^{2}(\widehat{W}_{t},\mathbb{Q})$. ∎

Lemma 23**.**

There is an identity of mixed Hodge structures:

[TABLE]

Proof.

Note that the two functors $\phi_{\star}$ and $\phi_{!}$ concide, since the map $\phi$ is proper. Consider the Leray spectral sequence of the morphism $\phi$ with second page $E_{2}^{p,q}=H^{p}_{c}(R^{q}\phi_{\star}\mathbb{Q})\Longrightarrow H_{c}^{p+q}(\widehat{W},\mathbb{Q})$, which is known to degenerate at the second page. The groups contributing to $H^{3}_{c}(\widehat{W},\mathbb{Q})$ are:

[TABLE]

Identity (41) holds if and only if $H^{0}_{c}(\mathbb{C}^{\times},R^{3}{\phi}_{\star}\mathbb{Q})=H^{2}_{c}(\mathbb{C}^{\times},R^{1}{\phi}_{\star}\mathbb{Q})=(0)$, and indeed this is so, by Lemma 22$(7)$. ∎

Lemma 24**.**

There is an isomorphism of mixed sheaves on $\mathbb{C}^{\times}$:

[TABLE]

Proof.

The sketch of the proof is as follow: let $\Omega\subset\mathbb{C}^{\times}$ and $j\colon\Omega\hookrightarrow\mathbb{C}^{\times}$ be as in Lemma 22 and let $p_{\Omega}\colon Y_{\Omega}=p^{-1}(\Omega)\to\Omega$ be the induced morphism; we first construct a homomorphism

[TABLE]

and then we show that it is an isomorphism. This concludes the proof since on the one hand it is obvious that $j_{\star}p_{\Omega\,\star}\mathbb{Q}_{Y_{\Omega}}=p_{\mathbb{C}^{\times}\star}\mathbb{Q}_{Y_{\mathbb{C}^{\times}}}$, on the other hand by Lemma 22(8) $j_{\star}R^{2}\phi_{\Omega\,\star}\mathbb{Q}_{\widehat{W}_{\Omega}}=R^{2}\phi_{\star}\mathbb{Q}_{\widehat{W}}$.

Below we construct a cycle

[TABLE]

inducing the homomorphism $z$ via the natural maps:

[TABLE]

Consider the diagram

[TABLE]

Let $\overline{Z}\subset\widehat{W}_{\Omega}$ be the union over all $t\in\Omega$ of the four curves $C_{t,1},\dots,C_{t,4}$ in the fibre $\widehat{W}_{t}$ described in Lemma 22, that is:

[TABLE]

We have that:

[TABLE]

is the sum of two irreducible components and we take $Z$ to be one of these components. In order to see this, let $K=\mathbb{C}(t)$ and $L=K(\sqrt{\delta_{1}},\sqrt{\delta_{2}})$, as in the proof of Lemma 22, and consider the field extensions $K\subset K(\sqrt{\delta})\subset L$. The key point is to notice that $Y_{\Omega}$ is a $2:1$ cover of $\Omega$ with function field $k(Y_{\Omega})=K(\sqrt{\delta})$. Indeed, by (23) $Y$ is a $2:1$ cover of $\mathbb{P}^{1}$ branched at the $6k+4$ roots of the polynomial

[TABLE]

By (33) this is the polynomial $\delta(t)=\delta_{1}(t)\cdot\delta_{2}(t)$. Then, since the generator of the Galois group $\mathrm{Gal}(L/K(\sqrt{\delta}))$ exchanges $C_{1}$ with $C_{4}$ and $C_{2}$ with $C_{3}$ on $\widehat{W}_{L}$, by Galois descent the cycle $C_{1}+C_{2}+C_{3}+C_{4}$ on $\widehat{W}_{k(Y_{\Omega})}$ splits into two components $C_{1}+C_{4}$ and $C_{2}+C_{3}$, each defined over $k(Y_{\Omega})$ and corresponding to the two irreducible components $Z_{1}$ and $Z_{2}$. Note also that $p^{\prime}_{\Omega\,\star}Z=\overline{Z}$.

It remains to show that the induced homomorphism $z$ is an isomorphism. This can be checked at the generic point $\eta$, or indeed at any point $t\in\Omega$. This is precisely the statement in Lemma 22$(4)$ that the set $\{C_{1}+C_{4},C_{2}+C_{3}\}$ is a basis of $\operatorname{Pic}(\widehat{W}_{t})_{\mathbb{Q}}=H^{2}(\widehat{W}_{t},\mathbb{Q})$. ∎

Appendix A Toric MMP

Let $W$ and $\widehat{W}$ be the fibre of (22) over $a$ and its partial compactification (28) in $G=\mathbb{P}(1,1,2,2)/_{\mu_{2}}\times\mathbb{C}^{\times}$, as in Section (5).

In this Appendix, we explain how we discovered the del Pezzo fibration $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$, and the substitution (26), by the methods of the minimal model program for toric hypersurfaces.

The space $\widehat{W}$ has canonical but not terminal singularities. In Section A.3 we construct a birational model $W^{\prime}$ with terminal singularities and a Mori fibration $\psi\colon W^{\prime}\to\mathbb{F}_{1}$.

As in Section (5), we fix once and for all an integer $k>0$ and a value $a\neq a_{k,0}$, and we omit all reference to $k$ and $a$ in what follows.

Notation

We set up our notation for toric varieties. For a lattice $L$, we denote by $L_{\mathbb{R}}=L\otimes_{\mathbb{Z}}\mathbb{R}$ the associated real vector space. For a torus $\mathbb{T}$, we denote by $M=\mathrm{Hom}(\mathbb{T},\mathbb{C}^{\times})$ the character lattice and let $N=\mathrm{Hom}(M,\mathbb{Z})$.

For a fan $\Sigma\subset N_{R}$, we denote by $F_{\Sigma}$ the associated toric variety. We denote by $\rho_{i}$ the primitive generators of the $1$-dimensional cones of $\Sigma$, and by $D_{i}\subset F_{\Sigma}$ the corresponding divisors.

If $F$ is a proper toric variety and $D=\sum a_{i}D_{i}$ is a Weil divisor on $F$, then we denote by

[TABLE]

the polytope of $D$; it is well known that $P_{D}\cap M$ is a basis of $H^{0}(F,D)$.

Given a rational polytope $P\subset M_{\mathbb{R}}$, we denote by $\Sigma_{P}$ the normal fan of $P$. If $P$ is not full dimensional $\Sigma_{P}$ is a generalised fan, that is, all the cones of $\Sigma_{P}$ contain a fixed vector subspace $\sigma_{0}$ with associated lattice $N_{0}=N\cap\sigma_{0}$. We denote by $F_{P}$ the toric variety for the fan $\Sigma_{P}/\sigma_{0}\subset(N/N_{0})_{\mathbb{R}}$.

We recall two well-known facts about toric varieties which we use repeatedly:

Fact 1

There is a 1-to-1 correspondence between:

1. (I)

the set of polarised toric varieties $(F,D)$, that is, pairs of a proper toric variety $F$ and (torus invariant) ample divisor $D$, and 2. (II)

the set of full-dimensional rational polytopes $P\subset M_{\mathbb{R}}$.

Given a pair $(F,D)$, the corresponding polyhedron is $P=P_{D}$. Conversely, given $P$, we let $F=F_{P}$ and $D=\sum b_{i}D_{i}$ where $P=\bigcap_{i}\{m\colon\langle\rho_{i},m\rangle\geq-b_{i}\}$ is the unique facet presentation of $P$.

Fact 2

Suppose that $D$ is a nef divisor on a proper toric variety $F$. Let $P=P_{D}$ be the polytope of $D$ and $(F_{P},D_{P})$ the corresponding polarised pair. Then $(F_{P},D_{P})$ is an ample model of $(F,D)$, in other words:

1. (i)

There is a proper toric morphism $f\colon F\to F_{P}$, and 2. (ii)

$D=f^{\star}D_{P}$.

A.1 The toric variety associated to $W$

Let $M=\mathrm{Hom}(\mathbb{T}^{4},\mathbb{C}^{\times})$ be the group of characters of the torus $\mathbb{T}^{4}$ in Equation (7) and let $N$ be its dual lattice. Denote by $\{e_{1},e_{2},e_{3},e_{4}\}$ the basis of $N$ dual to the basis of $M$ consisting of the coordinate functions $u_{1},u_{2},u_{3},u_{4}$ on $\mathbb{T}^{4}$.

In this Section we study the Newton polytope $P\subset M_{\mathbb{R}}$ of the polynomial in Equation (22), defining the variety $W\subset\mathbb{T}^{4}$, and the compactification of $W$ in the toric variety $F=F_{P}$.

The polytope $P$ is a 4-dimensional lattice polytope with six vertices $u_{0}=\underline{0}$, $u_{i}$ for $i\in\{1,2,3,4\}$ and $u_{5}=(2,2k+1,2k+1,4k+1)$, and facet presentation:

[TABLE]

where $(\rho_{i},a_{i})\in N\times\mathbb{Z}$ are:

[TABLE]

Hence the fan $\Sigma=\Sigma_{P}$ is a complete fan with six $4$-dimensional cones and eight rays generated by the vectors $\rho_{i}$, and $F$ is a compact $4$-dimensional toric variety. Note that $P$ does not depend on $a$ and so neither do $\Sigma$ and $F$.

The compactification of $W$ in $F$ corresponds to an element of the linear system $|O_{F}(D)|,$ where $D=\sum_{i=1}^{8}a_{i}D_{i}$ is the ample divisor of $F$ associated to $P$ (as in Fact $1$ above).

Remark 25**.**

The variety $F$ is not $\mathbb{Q}$-Gorenstein. Let $F^{\prime}\to F$ be any small birational morphism such that $K_{F^{\prime}}$ is $\mathbb{Q}$-Cartier. Then $F^{\prime}$ has noncanonical singularities. Indeed, consider for instance the affine open subsets $F^{\prime}_{\tau_{1}}$ and $F^{\prime}_{\tau_{2}}$ corresponding to the cones

[TABLE]

of the fan of $F^{\prime}$. Then

[TABLE]

The key point here is that the surface quotient singularities $\frac{1}{4k+2}(1,1)$ and $\frac{1}{4k+1}(1,2k)$ are not canonical: the vectors $\frac{1}{4k+2}(1,1)$ and $\frac{1}{4k+1}(1,2k)$ correspond to valuations with discrepancies $-\frac{4k}{4k+2}$ and $-\frac{2k}{4k+1}$. The weighted blow ups with weights $\frac{1}{4k+2}(1,1)$ and $\frac{1}{4k+1}(1,2k)$ are the minimal canonical partial resolutions of these singularities.

Our aim is to construct a birational model for $F$ with canonical singularities, as final product of a variant of the minimal model program.

A.2 A Mori fibre space structure

The plan

In this Section our plan is to:

1. (i)

choose and construct a $\mathbb{Q}$-Gorenstein partial resolution $f\colon\widetilde{F}\to F$ of $F$; 2. (ii)

run the minimal model program with scaling for $(\widetilde{F},\widetilde{D})$, where $\widetilde{D}=f^{\star}(D)$; 3. (iii)

study the singularities of the final product $(\overline{F},\overline{D})$ and the equation of $W$ in $\overline{F}$.

A.2.1 The $\mathbb{Q}$-Gorenstein partial resolution

Set $\rho_{9}=\frac{1}{4k+2}(\rho_{5}+\rho_{8})=(0,-1,-1,1)$ and $a_{9}=1$. (For $i\in\{1,\dots,8\}$ let $a_{i}$ be as in (44).) For $\varepsilon\geq 0$ consider the polytope:

[TABLE]

For $0<\varepsilon\ll 1$, $P(\varepsilon)$ is $4$-dimensional and the presentation in Equation (45) is the facet presentation of $P(\varepsilon)$, and the normal fan of $P(\varepsilon)$ is independent of $\varepsilon$; denote it by $\widetilde{\Sigma}$ and let $\widetilde{F}=F_{P(\varepsilon)}$ be the corresponding toric variety. Let $\widetilde{D}=\sum_{i=1}^{9}a_{i}D_{i}$; by Fact 1 above the divisor

[TABLE]

is ample on $\widetilde{F}$.

Lemma 26**.**

1. (a)

$\widetilde{F}$* is $\mathbb{Q}$-Gorenstein and the obvious map $f\colon\widetilde{F}\dasharrow F$ is a morphism;* 2. (b)

$\widetilde{D}=f^{\star}(D)$.

Sketch of proof.

Note that $\widetilde{D}$ is nef and $\widetilde{D}=\widetilde{D}(0)$. For $\varepsilon=0$ we have that

[TABLE]

where the second equality above follows from the choice $a_{9}=1$; thus by Fact 2 above $(F,D)$ is the ample model of $(\widetilde{F},\widetilde{D})$: in other words there is a morphism $f\colon\widetilde{F}\to F$ and $\widetilde{D}=f^{\star}(D_{P})$. Since for $0<\varepsilon\ll 1$ the divisor $\widetilde{D}(\varepsilon)=\varepsilon K_{\widetilde{F}}+\widetilde{D}$ is ample on $\widetilde{F}$, $K_{\widetilde{F}}\sim_{f}1/\varepsilon\widetilde{D}(\varepsilon)$ is $f$-ample. Finally, one can check explicitly that $\widetilde{F}$ is $\mathbb{Q}$-Gorenstein. ∎

A.2.2 The minimal model program

For $0<\varepsilon\ll 1$ the divisors $\widetilde{D}(\varepsilon)$ are ample on $\widetilde{F}$, since $K_{\tilde{F}}$ is $f$-ample and $D\subset F$ is ample. We recover $\widetilde{F}$ as the toric variety whose spanning fan is the normal fan of the polytopes $P(\epsilon)$ for small values of $0<\varepsilon$.

The minimal model program with scaling consists of an inductively defined finite sequence of toric varieties $F_{j}$, divisors $D_{j}$, and a strictly increasing sequence of rational numbers $\varepsilon_{j}$, and birational maps:

[TABLE]

where:

We start with $F_{0}=\widetilde{F}$, $D_{0}=\widetilde{D}$, and \varepsilon_{0}=\max\{\varepsilon>0\mid\varepsilon K_{\widetilde{F}}+\widetilde{D}\;\text{is nef on \widetilde{F}}\}; 2. 2.

For $j\geq 0$, $t_{j}\colon F_{j}\dasharrow F_{j+1}$ is the divisorial contraction or flip of the face $R_{j}\subset\mathrm{NE}(F_{j})$ with $(\varepsilon_{j}K_{j}+D_{j})_{|R_{j}}=0$ and $(K_{j}+D_{j})_{|R_{j}}<0$; 3. 3.

For $j>0$ $D_{j}$ is the proper transform of $D_{j-1}$ on $F_{j}$, $K_{j}=K_{F_{j}}$, and:

[TABLE] 4. 4.

The program ends at $F_{r}=\overline{F}$ where either:

The pair $(\overline{F},\overline{D})$ is a minimal model

that is $\overline{\varepsilon}=1$ and $K_{\overline{F}}+\overline{D}$ is nef; or

The pair $(\overline{F},\overline{D})$ is a Mori fibre space (Mfs)

that is $\overline{\varepsilon}<1$, the contraction $\psi\colon\overline{F}\to S$ of $R_{r}$ has relative dimension $>0$ and $K_{\overline{F}}+\overline{D}$ is $\psi$-ample. In this case $\overline{\varepsilon}$ is the quasi-effective threshold of the pair $(\widetilde{F},\widetilde{D})$:

[TABLE]

By Fact 1, we recover $F_{j}$ as the toric variety whose spanning fan is the normal fan of the polytopes $P(\varepsilon)$ for $\varepsilon\in(\varepsilon_{j-1},\varepsilon_{j})$, since the divisors $\varepsilon K_{j}+D_{j}$ are ample on $F_{j}$; the threshold values $\cdots<\varepsilon_{j}<\varepsilon_{j+1}<\cdots$ are those where the polytope $P(\varepsilon)$ changes shape.

A.2.3 The final product

Returning to the polytopes of Equation (45), $P(1)=\emptyset$, thus the minimal model program just described will end with a Mfs. The sequence of threshold values is777We computed the threshold values as follows. Consider the $5$-dimensional lattice $N^{\prime}=N\oplus\mathbb{Z}$ and the dual lattice $M^{\prime}$, whose elements are pairs $(m,r)$ with $m\in M$ and $r\in\mathbb{Z}$; also denote by $q\colon M^{\prime}\to\mathbb{Z}$ the projection to the second factor. Let $\rho^{\prime}_{i}=(\rho_{i},-1)\in N^{\prime}$, $i=1,\dots,10$ and let $P^{\prime}$ be the $5$-dimensional polyhedron:

$P^{\prime}=\bigcap_{i=1}^{10}\{m^{\prime}\in{(M^{\prime})}_{\mathbb{R}}:\langle m^{\prime},\rho^{\prime}_{i}\rangle\geq-a_{i}\}\subset{(M^{\prime})}_{\mathbb{R}}$

Then $\forall\varepsilon$ $P(\varepsilon)=q_{\mathbb{R}}^{-1}(\varepsilon)\cap P^{\prime}$ and the threshold values occur at the vertices of $P^{\prime}$. :

[TABLE]

Thus $\overline{\varepsilon}=2/3$,

[TABLE]

and $S=\mathbb{P}^{1}$. More precisely, the normal fan of $P\left(2/3\right)$ has two maximal cones:

[TABLE]

intersecting along the hyperplane $C_{0}=\{n\in N_{\mathbb{R}}:\ n_{2}+n_{3}+2n_{4}=0\}\subset N_{\mathbb{R}}$.

For all $\varepsilon\in\left((2k+1)/(3k+2),2/3\right)$, $P(\varepsilon)$ is a $4$-dimensional polytope with $9$ vertices and $7$ facets, whose normals are generated by the vectors $\rho_{1},\rho_{2},\rho_{3},\rho_{5},\rho_{6},\rho_{7},\rho_{10}$; thus its normal fan $\overline{\Sigma}$ has $7$ rays and $9$ maximal cones, which are listed in Figure 6. One can check that the $4$-dimensional toric variety $\overline{F}$ is $\mathbb{Q}$-Gorenstein but not $\mathbb{Q}$-factorial, and has canonical singularities.

The toric morphism $\psi\colon\overline{F}\to\mathbb{P}^{1}$ is induced by the quotient map $N\to N/N_{0}$, where $N_{0}=N\cap C_{0}$, sending

[TABLE]

The preimage via $\psi\colon\overline{F}\to\mathbb{P}^{1}$ of the torus $\mathbb{C}^{\times}\subset\mathbb{P}^{1}$ is the non-compact toric variety $G$ associated to the subfan $\Delta\subset\overline{\Sigma}$ given by the cones of $\overline{\Sigma}$ contained in $C_{0}$. The fan $\Delta$ has four $3$-dimensional cones (see Figure 6):

[TABLE]

Since the quotient $N/N_{0}$ is torsion-free, we may choose a splitting $N=N_{0}\oplus N_{1}$ and regard $\Delta$ as a fan $\Delta_{0}$ in $N_{0}$; we have $\Delta=\Delta_{0}\oplus\{0\}$, thus $G=G_{0}\times\mathbb{C}^{\times}$, where $G_{0}$ is the $3$-dimensional toric variety with spanning fan $\Delta_{0}\subset N_{0}$. To determine $G_{0}$, note that the vectors $\rho_{1},\rho_{6},\rho_{7},\rho_{10}$ spanning the hyperplane $C_{0}$ satisfy the linear relation: $\rho_{6}+\rho_{7}+2\rho_{1}+2\rho_{10}=0$. However $G_{0}$ is not the weighted projective space $\mathbb{P}(1,1,2,2)$, as, for instance, $1/2\cdot(\rho_{6}+\rho_{1}+\rho_{10})\in N_{0}$. Then write:

[TABLE]

We have $M_{00}\supset M_{0}=\{m\in M_{00}:\ m_{1}+m_{2}+m_{3}=0\ \mathrm{mod}\ 2\}$ and $\mathbb{C}[M_{0}]$ is the ring of invariants ${\mathbb{C}[M_{00}]}^{\mu_{2}}$, where the group where $\mu_{2}$ acts on $\mathbb{C}[M_{00}]$ by $\xi\cdot\chi^{m}=\xi^{m_{1}+m_{2}+m_{3}}\chi^{m}$. Equivalently $G_{0}$ is the quotient variety:

[TABLE]

Denoting by $x_{1},x_{2},y_{1},y_{2}$ the weighted homogeneous coordinates of $\mathbb{P}(1,1,2,2)$, the $\mu_{2}$ action has weights $1/2(1,1,1)$ on the two smooth charts $\{x_{1}=1\}$ and $\{x_{2}=1\}$, and $1/4(1,-1,0)$ on the two singular charts $\{y_{1}=1\}$ and $\{y_{2}=1\}$. Thus $G$ is the toric variety of Equation (25).

To determine the equation of $W\subset\mathbb{T}^{4}$ in $G=\mathbb{P}(1,1,2,2)/\mu_{2}\times\mathbb{C}^{\times}$, we choose the basis of $N_{\mathbb{R}}$ given by the four vectors $\rho_{6},\rho_{1},\rho_{10},e_{2}$ and denote the dual basis by $x$, $y$, $z$, $t$. In terms of $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$:

[TABLE]

hence

[TABLE]

This is the substitution (26), thus $W$ identifies with the $3$-fold of Equation (27) in $\mathbb{T}_{G}$. The del Pezzo fibration $\phi\colon\widehat{W}\to\mathbb{C}^{\times}$ of Section 5 is the restriction of the toric morphism $\varphi\colon\overline{F}\to\mathbb{P}^{1}$.

A.3 A conic bundle structure

The del Pezzo fibration $\phi\colon\widehat{W}\to\operatorname{\mathbb{C}^{\times}}$ is not a Mori fibre space in the Mori category: $\widehat{W}$ has strictly canonical singularities along two sections of $\phi$ corresponding to the orbifold points $p_{t}^{\pm}\in\widehat{W}_{t}$ (all $t$) of Lemma 22(3).

In this Section we construct a birational model $W^{\prime}$ with terminal singularities and a Mori fibration $\psi\colon W^{\prime}\to\mathbb{F}_{1}$ in the Mori category. In our view, $\widehat{W}$ is a better model in which to find the group $H^{1}(Y,\mathbb{Q})$.

Construction

Consider the toric variety $G^{\prime}$ with weight matrix888This matrix defines an action of $\mathbb{G}_{m}^{3}$ on $\mathbb{A}^{7}$ and $G^{\prime}$ is a GIT quotient.:

[TABLE]

and irrelevant ideal $(t_{1},t_{2})(v_{1},v_{2})(z,z_{1},z_{2})$. Note the morphism $G^{\prime}\to\mathbb{F}_{1}$. Set

[TABLE]

and consider the hypersurface $W^{\prime}\subset G^{\prime}$ given by

[TABLE]

Theorem 27**.**

1. (A)

$W^{\prime}$* has terminal singularities and the morphism $\psi\colon W^{\prime}\to\mathbb{F}_{1}$ is a conic bundle Mfs;* 2. (B)

There is a commutative diagram:

[TABLE]

Proof.

(A) It is possible, and not too hard, to verify explicitly that $W^{\prime}$ has isolated cA singularities.

(B) We start by exibiting an explicit birational map from a chart of $G$ to a chart of $G^{\prime}$.

Consider the chart $(t_{2}=v_{2}=z=1)\subset G^{\prime}$; this is isomorphic to $\mathbb{C}^{4}$ with coordinate functions $t_{1},v_{1},z_{1},z_{2}$. In this chart, $W^{\prime}$ is given by the equation:

[TABLE]

Next consider the chart $(x_{2}=1)\subset G$; this is isomorphic to $\mathbb{G}_{m}\times\frac{1}{2}(1,1,1)$ with coordinate ring

[TABLE]

generated by the functions $t,x_{1}^{2},x_{1}y_{1}$, &c. In this chart, $\widehat{W}$ is given by the equation:

[TABLE]

We define a rational map from the chart in $G$ to the chart in $G^{\prime}$ by:

[TABLE]

This map is in fact birational with inverse given by:

[TABLE]

It is easy to see that the equation for $\widehat{W}$ is transformed into the equation for $W^{\prime}$. ∎

Remark 28**.**

The variety $W^{\prime}$ has many singular points (more precisely, above $\widetilde{\delta}_{1}=v_{j}=0$, $t_{1}=v_{1}=0$, $t_{2}=v_{j}=0$, $j=1,2$); thus the conic bundle $\psi\colon W^{\prime}\to\mathbb{F}_{1}$, although it is a Mori fibre space, is not standard (in the birational geometry literature, a 3-fold Mfs $f\colon X\to S$ is a standard conic bundle if $\dim S=2$ and $X$ is nonsingular, which implies that $S$ is also nonsingular, all fibres are conics, and the discriminant is normal crossing). Hence the conjectural rationality criteria [16, 14, 15, 13] do not directly apply, though they suggest that $W^{\prime}$ is nonrational (and possibly even birationally rigid). It would be interesting to study the geometry of $W^{\prime}$ further.

Remark 29**.**

The conic bundle $\psi\colon W^{\prime}\to\mathbb{F}_{1}$ is an analog of a construction that in the singularity theory and mirror symmetry literature is called a “double suspension.” It is possible, of course, to find $H^{1}(Y,\mathbb{Q})$ in $H^{3}(W^{\prime},\mathbb{Q})$, but it is easier to find it in $H^{3}(\widehat{W},\mathbb{Q})$.