Intrinsicness of the Newton polygon for smooth curves on ${\mathbb {P}}^1\times {\mathbb {P}}^1$

Abstract

Let C be a smooth projective curve in ${\mathbb {P}}^1 \times {\mathbb {P}}^1$ of genus $g\ne 4$, and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon $\Delta$. Then we show that the convex hull $\Delta ^{(1)}$ of the interior lattice points of $\Delta$ is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of $\Delta$-non-degenerate curves are encoded in the combinatorics of $\Delta ^{(1)}$, if $\Delta$ satisfies some mild conditions.

1 Introduction

Let $f\in k[x^{\pm 1},y^{\pm 1}]$ be an irreducible Laurent polynomial over an algebraically closed field k of characteristic zero and let U(f) be the curve it defines in the two-dimensional torus ${\mathbb {T}}^2=(k^*)^2$. The Newton polygon $\Delta =\Delta (f)$ of f is the convex hull in ${\mathbb {R}}^2$ of all the exponent vectors in ${\mathbb {Z}}^2$ of the monomials that appear in f with a non-zero coefficient. We will always assume that $\Delta$ is two-dimensional. We say that f is non-degenerate with respect to its Newton polygon $\Delta$ (or more briefly, f is $\Delta$-non-degenerate) if and only if for each face $\tau \subset \Delta$ (including $\tau =\Delta$) the system

$$\begin{aligned} f_\tau =\frac{\partial f_\tau }{\partial x}=\frac{\partial f_\tau }{\partial y}=0 \end{aligned}$$

does not have any solutions in ${\mathbb {T}}^2$. Here, $f_\tau$ is obtained from f by only considering the terms that are supported on $\tau$. This condition is generically satisfied. Consider the map

$$\begin{aligned} \varphi _\Delta : {\mathbb {T}}^2 \rightarrow {\mathbb {P}}^{\sharp (\Delta \cap {\mathbb {Z}}^2)-1} : (x,y) \mapsto (x^i y^j)_{(i,j)\in \Delta \cap {\mathbb {Z}}^2}. \end{aligned}$$

The Zariski closure of its full image $\varphi _\Delta ({\mathbb {T}}^2)$ is a toric surface $\text {Tor}(\Delta )$, while the Zariski closure of $\varphi _\Delta (U(f))$ is a hyperplane section $C_f$ of $\text {Tor}(\Delta )$, which is smooth if f is non-degenerate. We will denote the projective coordinates of ${\mathbb {P}}^{\sharp (\Delta \cap {\mathbb {Z}}^2)-1}$ by $X_{i,j}$ where (i, j) runs over $\Delta \cap {\mathbb {Z}}^2$.

We say that a smooth curve C is $\Delta$-non-degenerate if and only if it is birationally equivalent to U(f) for a $\Delta$-non-degenerate Laurent polynomial f. Note that if C is moreover projective, then it is isomorphic to $C_f$. If C is $\Delta$-non-degenerate, then a lot of its geometric properties are encoded in the combinatorics of the lattice polygon $\Delta$. For instance, its geometric genus g(C) equals the number of interior lattice points of $\Delta$ [8]. Similar interpretations were recently provided for the gonality [3, 7], the Clifford index and dimension [3, 7], the scrollar invariants associated to a gonality pencil [3] and Schreyer’s tetragonal invariants [5].

Given this long list, the following question (initiated in [5]) naturally arises: to what extent can we recover $\Delta$ from the geometry of a $\Delta$-non-degenerate curve? At least, we have to allow two relaxations to this question. First, we can only expect to find back the polygon $\Delta$ up to a unimodular transformation, i.e. an affine map of the form

$$\begin{aligned} \chi : {\mathbb {R}}^2\rightarrow {\mathbb {R}}^2 : \begin{pmatrix} x \\ y \end{pmatrix} \mapsto A \begin{pmatrix} x \\ y \end{pmatrix} + B \end{aligned}$$

with $A\in \text {GL}_2({\mathbb {Z}})$ and $B\in {\mathbb {Z}}^2$, since these maps correspond to automorphisms of ${\mathbb {T}}^2$. Secondly, we can (usually) only hope to recover the convex hull of the interior lattice points of $\Delta$, denoted by $\Delta ^{(1)}$ (see [5] for an easy example demonstrating the need for this relaxation). In fact, all the aforementioned combinatorial interpretations are in terms of the combinatorics of $\Delta ^{(1)}$ rather than $\Delta$ (e.g. $g(C)=\sharp (\Delta ^{(1)}\cap {\mathbb {Z}}^2)$).

Given a $\Delta$-non-degenerate curve C, we say that the Newton polygon $\Delta$ is intrinsic for C if and only if for all $\Delta '$-non-degenerate curves $C'$ that are birationally equivalent to C, we have that $\Delta ^{(1)}\cong \Delta '^{(1)}$. Hereby, we use $\cong$ to denote the unimodular equivalence relation. Before stating some intrinsicness results, we give notations for some special lattice polygons:

$$\begin{aligned} \square _{a,b}&= \text {conv}\{(0,0),(a,0),(0,b),(a,b)\} \text { for } a,b\in {\mathbb {Z}}_{\ge 0},\\ \Sigma&= \text {conv}\{(0,0),(1,0),(0,1)\}, \\ \Upsilon&= \text {conv}\{(-1,-1),(1,0),(0,1)\}. \end{aligned}$$

The Newton polygon is intrinsic for all rational ($\Delta ^{(1)}=\emptyset$), hyperelliptic ($\Delta ^{(1)}$ is one-dimensional, and therefore determined by the genus) and trigonal curves of genus at least 5 ($\Delta ^{(1)}$ has lattice width 1, and is determined by the Maroni invariants). However, there are trigonal curves of genus 4 for which $\Delta$ is not intrinsic: there exist curves which are non-degenerate with respect to polygons $\Delta$ and $\Delta '$, with $\Delta ^{(1)}=\Upsilon$ and $\Delta '^{(1)}=\square _{1,1}$. Intrinsicness of the Newton polygon for tetragonal curves was studied in [5]: the Newton polygon $\Delta$ is intrinsic if $g(C)\bmod 4\in \{2,3\}$, but it might occasionally be not intrinsic if $g(C)\bmod 4\in \{0,1\}$. From [3], it follows that non-degenerate smooth plane curves of degree $d\ge 3$ ($\Delta ^{(1)}\cong (d-3)\Sigma$) and curves with Clifford dimension 3 ($\Delta ^{(1)}\cong 2\Upsilon$) have an intrinsic Newton polygon. Moreover, a partial result was given for non-degenerate curves on Hirzebruch surfaces ${\mathcal {H}}_n$: the value n is intrinsic.

In this paper, we examine intrinsicness of $\Delta$ for curves on ${\mathbb {P}}^1\times {\mathbb {P}}^1$. Namely, we will show that a $\Delta$-non-degenerate curve C of genus $g\ne 4$ can be embedded in ${\mathbb {P}}^1\times {\mathbb {P}}^1$ (if and) only if $\Delta ^{(1)}=\emptyset$ or $\Delta ^{(1)}\cong \square _{a,b}$ for $a,b\in {\mathbb {Z}}_{\ge 0}$ satisfying $g=(a+1)(b+1)$; see Theorem 18 in Sect. 3. In order to prove this result, we give a combinatorial interpretation for the first scrollar Betti numbers of $\Delta$-non-degenerate curves with respect to a gonality pencil, as soon as $\Delta$ satisfies some mild conditions (see Sect. 2).

Notations Let ${\mathbb {P}}^N$ be a projective space with coordinates $(X_0:\ldots :X_N)$. For each projective variety $V\subset {\mathbb {P}}^N$, we write ${\mathcal {I}}(V)\subset k[X_0,\ldots ,X_N]$ to indicate the homogeneous ideal of V and ${\mathcal {I}}_d(V)\subset {\mathcal {I}}(V)$ to indicate its homogeneous degree d piece. If $J\subset k[X_0,\ldots ,X_N]$ is a homogeneous ideal, then ${\mathcal {Z}}(J)\subset {\mathbb {P}}^N$ is the zero locus of the polynomials in J.

2 First scrollar Betti numbers

2.1 Definition

We start by recalling the definition and some properties of rational normal scrolls.

Let $n \in {\mathbb {Z}}_{\ge 2}$ and let ${\mathcal {E}}={\mathcal {O}}(e_1)\oplus \cdots \oplus {\mathcal {O}}(e_n)$ be a locally free sheaf of rank n on ${\mathbb {P}}^1$. Denote by $\pi :{\mathbb {P}}({\mathcal {E}})\rightarrow {\mathbb {P}}^1$ the corresponding ${\mathbb {P}}^{n-1}$-bundle. We assume that $0\le e_1\le e_2\le \cdots \le e_n$ and that $e_1 + e_2 + \dots + e_n \ge 2$. Set $N=e_1+e_2+\cdots +e_n+n-1$. Then the image $S=S(e_1,\ldots ,e_n)$ of the induced morphism

$$\begin{aligned} \mu :{\mathbb {P}}({\mathcal {E}})\rightarrow {\mathbb {P}}H^0({\mathbb {P}}({\mathcal {E}}),{\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(1)), \end{aligned}$$

when composed with an isomorphism ${\mathbb {P}}H^0({\mathbb {P}}({\mathcal {E}}),{\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(1)) \rightarrow {\mathbb {P}}^N$, is called a rational normal scroll of type $(e_1, \dots , e_n)$. Up to automorphisms of ${\mathbb {P}}^N$, rational normal scrolls are fully characterized by their type.

They can also be described in a geometric way: consider linearly independent projective subspaces ${\mathbb {P}}_1^{e_1},\ldots ,{\mathbb {P}}_n^{e_n}\subset {\mathbb {P}}^N$ of dimensions $e_1,\ldots ,e_n$, so their span is the whole projective space ${\mathbb {P}}^N$. For each $i\in \{1,\ldots ,n\}$, fix a rational normal curve in ${\mathbb {P}}_i$ of degree $e_i$ parametrized by a Veronese map $\nu _i:{\mathbb {P}}^1\rightarrow {\mathbb {P}}_i^{e_i}$. Then

$$\begin{aligned} S=\cup _{P\in {\mathbb {P}}^1} \langle \nu _1(P),\ldots ,\nu _n(P)\rangle \subset {\mathbb {P}}^N \end{aligned}$$

is a rational normal scroll of type $(e_1,\ldots ,e_n)$.

The scroll is smooth if and only if $e_1>0$. In this case, $\mu : {\mathbb {P}}({\mathcal {E}}) \rightarrow S$ is an isomorphism. If $0=e_1=\cdots =e_\ell <e_{\ell +1}$ with $1 \le \ell <n$, then the scroll is a cone with an $(\ell -1)$-dimensional vertex. In this case $\mu : {\mathbb {P}}({\mathcal {E}}) \rightarrow S$ is a resolution of singularities and

$$\begin{aligned} \mu _\lambda : {\mathbb {P}}({\mathcal {E}})\cong {\mathbb {P}}({\mathcal {E}}\otimes {\mathcal {O}}_{{\mathbb {P}}^1}(\lambda )) \rightarrow S'=S(e_1+\lambda ,\ldots ,e_n+\lambda ) \end{aligned}$$

is an isomorphism for all integers $\lambda >0$.

The Picard group of ${\mathbb {P}}({\mathcal {E}})$ is freely generated by the class H of a hyperplane section (more precisely, the class corresponding to $\mu ^*{\mathcal {O}}_{{\mathbb {P}}^N}(1)$) and the class R of a fiber of $\pi$; i.e.

$$\begin{aligned} \text {Pic}({\mathbb {P}}({\mathcal {E}}))={\mathbb {Z}}H \oplus {\mathbb {Z}}R. \end{aligned}$$

We have the following intersection products:

$$\begin{aligned} H^n=e_1+\cdots +e_n,\ H^{n-1}R=1 \ \text { and }\ R^2=0 \end{aligned}$$

(where $R^2 = 0$ means that any appearance of $R^2$ annihilates the product). If we denote the class which corresponds to $\mu _\lambda ^*{\mathcal {O}}_{{\mathbb {P}}^{N+n\lambda }}(1)$ by $H'$, we obtain the equality $H'=H+\lambda R$ in $\text {Pic}({\mathbb {P}}({\mathcal {E}}))$.

Let C / k be a smooth projective curve of genus g and gonality $\gamma \ge 4$. Assume that C is canonically embedded in ${\mathbb {P}}^{g-1}$ and fix a gonality pencil $g^1_\gamma$ on C. By [6, Thm. 2],

$$\begin{aligned} S=\bigcup _{D\in g^1_{\gamma }}\,\langle D\rangle \subset {\mathbb {P}}^{g-1} \end{aligned}$$

is a $(\gamma -1)$-dimensional rational normal scroll containing C. If S is of type $(e_1,\ldots ,e_{\gamma -1})$, the numbers $e_1,\ldots ,e_{\gamma -1}$ are called the scrollar invariants of C with respect to $g_{\gamma }^1$. Using the Riemann-Roch theorem, one can see that $e_{\gamma -1}\le \frac{2g-2}{\gamma }$.

The following theorem extends a result from [9] on tetragonal and pentagonal curves to arbitrary curves.

Theorem 1

Let C be a canonically embedded smooth projective curve of genus g and gonality $\gamma \ge 4$. If $g^1_\gamma$ is a gonality pencil on C, let $S\subset {\mathbb {P}}^{g-1}$ be the rational normal scroll swept out by $g^1_\gamma$ and let $C'$ be the strict transform of C under the resolution $\mu :{\mathbb {P}}({\mathcal {E}}) \rightarrow S$. Then there exist effective divisors $D_1,\ldots ,D_{(\gamma ^2-3\gamma )/2}$ on ${\mathbb {P}}({\mathcal {E}})$ along with integers $b_1, \dots , b_{(\gamma ^2-3\gamma )/2}$, such that $D_\ell \sim 2H-b_\ell R$ for all $\ell$ and $C'$ is the (scheme-theoretical) intersection of the $D_\ell$’s. Moreover, the multiset $\{b_1,\ldots ,b_{(\gamma ^2-3\gamma )/2}\}$ does not depend on the choice of the $D_\ell$’s, and

$$\begin{aligned} \sum _{\ell =1}^{(\gamma ^2-3\gamma )/2} b_\ell \, = \, (\gamma -3)(g-\gamma -1). \end{aligned}$$

Proof

Define $\beta _i=\frac{i(\gamma -2-i)}{\gamma -1}{\gamma \atopwithdelims ()i+1}$ and note that $\beta _1 = (\gamma ^2-3\gamma )/2$. The existence follows from [9, Cor. 4.4] and its proof, where the $D_\ell$’s come from an exact sequence of ${\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}$-modules

$$\begin{aligned} 0\rightarrow & {} {\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(-\gamma H+(g-\gamma +1)R)\rightarrow \sum _{\ell =1}^{\beta _{\gamma -3}} {\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(-(\gamma -2)H+b_\ell ^{(\gamma -3)}R)\rightarrow \cdots \nonumber \\\rightarrow & {} \sum _{\ell =1}^{\beta _2}{\mathcal {O}}_{{\mathbb {P}} ({\mathcal {E}})}(-3H+b_\ell ^{(2)}R) \rightarrow \sum _{\ell =1}^{\beta _1}{\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(-2H+b_\ell R)\rightarrow {\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}\rightarrow {\mathcal {O}}_{C'}\rightarrow 0.\nonumber \\ \end{aligned}$$

(1)

Tensoring (1) with ${\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(2H + bR)$ for a sufficiently large integer b and computing the Euler characteristics of the terms in the resulting exact sequence, one can show that

$$\begin{aligned} \sum _\ell b_\ell \, = \, (\gamma -3)(g-\gamma -1); \end{aligned}$$

see [1, Prop. 2.9].

We are left with showing the independence of the multiset $\{b_1,\ldots ,b_{(\gamma ^2-3\gamma )/2}\}$. Herefore, consider the exact sequence

$$\begin{aligned} \sum _{\ell = 1}^{\beta _1} {\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}(-D_\ell ) \rightarrow {\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})} \rightarrow {\mathcal {O}}_{C'} \rightarrow 0. \end{aligned}$$

(2)

If $\pi :{\mathbb {P}}({\mathcal {E}})\rightarrow {\mathbb {P}}^1$ is the ${\mathbb {P}}^{\gamma -2}$-bundle and $\xi$ is the generic point of ${\mathbb {P}}^1$, then

$$\begin{aligned} \pi ^{-1}(\xi )={\mathbb {P}}_{k(\xi )}^{\gamma -2}=\text {Proj}\, {\mathcal {S}}, \end{aligned}$$

where ${\mathcal {S}}=k(\xi )[x_0,\ldots ,x_{\gamma -2}]$. Applying $\cdot \,\otimes _{{\mathbb {P}}^1} k(\xi )$ to (2) yields an exact sequence of graded ${\mathcal {S}}$-modules, that can be extended to a minimal free resolution

$$\begin{aligned} 0 \rightarrow {\mathcal {S}}(-\gamma ) \rightarrow {\mathcal {S}}(-\gamma +2)^{\oplus \beta _{\gamma -3}} \rightarrow \cdots \rightarrow {\mathcal {S}}(-2)^{\oplus \beta _1}\rightarrow {\mathcal {S}}\rightarrow {\mathcal {S}}_{C'}\rightarrow 0 \end{aligned}$$

of the coordinate ring ${\mathcal {S}}_{C'}$ of $C'$ over $k(\xi )$ (see [9, Lemma 4.2] and [2, Step A of Thm. 2.1]). As explained in [9, proof of Thm. 3.2] and [2, proof of Step B of Thm. 2.1], this resolution can be lifted to a minimal free resolution of ${\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}$-modules extending (2). This resolution is unique up to isomorphism by [9, Thm. 3.2] or [2, Thm. 1.3], which implies the independence. $\square$

We call the invariants $b_1,\ldots ,b_{(\gamma ^2-3\gamma )/2}$ the first scrollar Betti numbers of C with respect to $g^1_\gamma$. The main goal of this section is to give a combinatorial interpretation for these invariants for non-degenerate curves.

In [5], we already treated the case of tetragonal $\Delta$-non-degenerate curves: the first scrollar Betti numbers are given by

$$\begin{aligned} \sharp (\partial \Delta ^{(1)}\cap {\mathbb {Z}}^2)-4 \quad \text {and} \quad \sharp (\Delta ^{(2)}\cap {\mathbb {Z}}^2)-1. \end{aligned}$$

These numbers are independent from the choice of the gonality pencil. This will no longer be true for non-degenerate curves of higher gonality.

2.2 Scrollar invariants for non-degenerate curves

Let f be a $\Delta$-non-degenerate Laurent polynomial and consider the corresponding smooth curve $C_f\subset \text {Tor}(\Delta )\subset {\mathbb {P}}^N$ with $N=\sharp (\Delta \cap {\mathbb {Z}}^2)-1$. Assume that the polygon $\Delta ^{(1)}$ is two-dimensional.

By [8], $C_f$ is a non-rational and non-hyperelliptic curve and there exists a canonical divisor $K_\Delta$ on $C_f$ such that

$$\begin{aligned} H^0(C_f,K_\Delta )=\langle x^iy^j\rangle _{(i,j)\in \Delta ^{(1)}\cap {\mathbb {Z}}^2} \end{aligned}$$

(where x, y are functions on $C_f$ through $\varphi _\Delta$). In particular, the curve $C_f$ has genus $g=\sharp (\Delta ^{(1)}\cap {\mathbb {Z}}^2)\ge 3$; see [3] for more details. Moreover, the Zariski closure $C=C_f^{can}$ of the image of U(f) under

$$\begin{aligned} \varphi _{\Delta ^{(1)}}:{\mathbb {T}}^2\hookrightarrow {\mathbb {P}}^{g-1}:(x,y)\mapsto (x^iy^j)_{(i,j)\in \Delta ^{(1)}\cap {\mathbb {Z}}^2} \end{aligned}$$

(3)

is a canonical model for $C_f$. We end up with the inclusions

$$\begin{aligned} C\subset T=\text {Tor}(\Delta ^{(1)})=\overline{\varphi _{\Delta ^{(1)}}({\mathbb {T}}^2)}\subset {\mathbb {P}}^{g-1}, \end{aligned}$$

where T is a toric surface since $\Delta ^{(1)}$ is two-dimensional.

A lattice direction is a primitive integer vector $v=(a,b)\in {\mathbb {Z}}^2$. The width $w(\Delta ,v)$ of $\Delta$ with respect to a lattice direction v is the smallest integer $\ell$ such that $\Delta$ is contained in the strip $k\le aY-bX\le k+\ell$ of ${\mathbb {R}}^2$ for some $k\in {\mathbb {Z}}$. The lattice width is defined as $\text {lw}(\Delta )=\min _v w(\Delta ,v)$. Lattice directions v that attain the minimum are called lattice-width directions.

In [3], we gave a combinatorial interpretation for the gonality $\gamma$ of $C=C_f^{can}$ (or $C_f$) in terms of the lattice width of $\Delta$:

$$\begin{aligned} \gamma ={\left\{ \begin{array}{ll} \text {lw}(\Delta )=\text {lw}(\Delta ^{(1)})+2 &{} \text {if } \Delta \not \cong 2\Upsilon \text { and } \Delta \not \cong d\Sigma \text { for all } d\in {\mathbb {Z}}_{\ge 4}, \\ \text {lw}(\Delta )-1=\text {lw}(\Delta ^{(1)})+2 &{} \text {if } \Delta \cong d\Sigma \text { for some } d\in {\mathbb {Z}}_{\ge 4}, \\ \text {lw}(\Delta )-1=\text {lw}(\Delta ^{(1)})+1 &{} \text {if } \Delta \cong 2\Upsilon , \end{array}\right. } \end{aligned}$$

where we use our assumption that $\Delta ^{(1)}$ is two-dimensional. From now on, we make the stronger assumption that $\gamma =\text {lw}(\Delta )\ge 4$, and that $\Delta ^{(1)}$ is not equivalent with $k\Sigma$ for any k or $\Upsilon$, hence $\Delta \not \cong d\Sigma$ and $\Delta \not \cong 2\Upsilon$. Then each lattice-width direction $v=(a,b)$ gives rise to a rational map

$$\begin{aligned} C\dashrightarrow {\mathbb {P}}^1: (x^iy^j)_{(i,j)\in \Delta ^{(1)}\cap {\mathbb {Z}}^2} \mapsto x^ay^b \end{aligned}$$

of degree equal to the gonality $\gamma$. We call the corresponding linear pencil $g_\gamma ^1$ of C a combinatorial gonality pencil. If $\Delta$ is sufficiently big (for a precise statement, see [3, Corollary 6.3]), each gonality pencil on C is combinatorial.

Fix a lattice-width direction v of $\Delta$. After applying a suitable unimodular transformation $\chi$, we may assume that $v=(1,0)$ and that $\Delta$ is contained in the horizontal strip $0\le Y\le \gamma$ in ${\mathbb {R}}^2$. So, the gonality map $C\dashrightarrow {\mathbb {P}}^1$ associated to v is the vertical projection to the x-axis. Write

$$\begin{aligned} i^{(-)}(j)=\min \{i\in {\mathbb {Z}}\,|\,(i,j)\in \Delta ^{(1)}\} \quad \text { and }\quad i^{(+)}(j)=\max \{i\in {\mathbb {Z}}\,|\,(i,j)\in \Delta ^{(1)}\} \end{aligned}$$

for all $j\in \{1,\ldots ,\gamma -1\}$. By [3, Theorem 9.1], the scrollar invariants $e_1,\ldots ,e_{\gamma -1}$ of C with respect to $g_\gamma ^1$ are equal to $E_j := i^{(+)}(j)-i^{(-)}(j)$ for $j\in \{1,\dots ,\gamma -1\}$ (up to order). In fact, a Zariski dense part of the scroll S is parametrized by

$$\begin{aligned} (a_1,\ldots ,a_{\gamma -1},x)\in {\mathbb {T}}^{\gamma }\mapsto & {} (a_j x^{i-i^{(-)}(j)})_{(i,j)\in \Delta ^{(1)}\cap {\mathbb {Z}}^2} \\= & {} (a_j,\ldots ,a_j x^{E_j})_{1\le j\le \gamma -1}\in {\mathbb {P}}^{g-1}. \end{aligned}$$

Note that $T=\text {Tor}(\Delta ^{(1)})\subset S$ since the map $\varphi _{\Delta ^{(1)}}$ can be obtained from the above parametrization by restricting to $a_j=x^{i^{(-)}(j)}y^j$, so we get the inclusions

$$\begin{aligned} C\subset T\subset S\subset {\mathbb {P}}^{g-1}. \end{aligned}$$

(4)

If S is singular, then $\mu : S'=S(e_1+\lambda ,\ldots ,e_{\gamma -1}+\lambda )\cong {\mathbb {P}}({\mathcal {E}}) \rightarrow S$ is a resolution of singularities for each integer $\lambda >0$ (hereby, we slightly abuse notation: $\mu$ is the map $\mu \circ \mu _\lambda ^{-1}$ using the notations in Sect. 2.1). Let $C'$ and $T'$ be the strict transforms of respectively C and T under $\mu$. For each lattice polygon $\Gamma \subset {\mathbb {R}}^2$, write $\Gamma [\lambda ]$ to denote the Minkowski sum of $\Gamma$ and $[(0,0),(\lambda ,0)]\subset {\mathbb {R}}^2$. In other words, $\Gamma [\lambda ]$ is obtained from $\Gamma$ by stretching it out in the horizontal direction over a distance $\lambda$. Using this notation, one can see that $T'=\text {Tor}(\Delta ^{(1)}[\lambda ])=\text {Tor}(\Delta [\lambda ]^{(1)})$. We end up with the inclusions

$$\begin{aligned} C'\subset T'\subset S'\subset {\mathbb {P}}^{g-1+\lambda (\gamma -1)}. \end{aligned}$$

(5)

2.3 First scrollar Betti numbers of toric surfaces

Let C be a $\Delta$-non-degenerate curve and fix a combinatorial gonality pencil $g^1_\gamma$ on C, corresponding to a lattice direction v. We work under the following assumptions:

• (i) $\Delta ^{(1)}$ is not equivalent with $k\Sigma$ for any k or $\Upsilon$, and $\gamma =\text {lw}(\Delta )\ge 4$,

• (ii) $v=(1,0)$ and $\Delta$ is contained in the horizontal strip $0\le Y\le \gamma$, so that $g^1_\gamma$ corresponds to the vertical projection,

• (iii) the curve C is canonically embedded, so that we obtain the sequence of inclusions $C \subset T \subset S \subset {\mathbb {P}}^{g-1}$ from (4).

Recall that the scrollar invariants $e_1, \dots , e_{\gamma - 1}$ of C with respect to $g^1_\gamma$ match with $E_1,\ldots ,E_{\gamma -1}$ (up to order). Consider $\mu : {\mathbb {P}}({\mathcal {E}}) \rightarrow S$ and let $T'\subset {\mathbb {P}}({\mathcal {E}})$ be the strict transform of $T=\text {Tor}(\Delta ^{(1)})$ under $\mu$, as in Sect. 2.2. If $\Delta$ satisfies the condition ${\mathcal {P}}_1(v)$ defined below (see Definition 4), we will provide effective divisors $D_1,\ldots ,D_{{\gamma -2 \atopwithdelims ()2}}$ on ${\mathbb {P}}({\mathcal {E}})$ along with integers $b_1, \dots , b_{{\gamma -2 \atopwithdelims ()2}}$, such that the following three conditions are satisfied:

• $T'$ is the (scheme-theoretical) intersection of the $D_\ell$’s,

• $D_\ell \sim 2H-b_\ell R$ for all $\ell$ (where H is the hyperplane class and R is the class of a fibre in $\text {Pic}({\mathbb {P}}({\mathcal {E}}))$), and,

• $\sum _{\ell =1}^{{\gamma -2 \atopwithdelims ()2}} b_\ell \, = (\gamma -4)g-(\gamma ^2-3\gamma )+\sharp (\partial \Delta ^{(1)}\cap {\mathbb {Z}}^2)$.

In what follows, we will also assume that $e_1 > 0$, so that ${\mathbb {P}}({\mathcal {E}})\cong S$. This condition is not essential (see Remark 9), but it allows us to work with the inclusion $T\subset S$ rather than $T'\subset {\mathbb {P}}({\mathcal {E}})$. For convenience, we will use the notation $D_{j_1,j_2}$ for the the divisors, where $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ such that $j_2-j_1\ge 2$, and denote the corresponding invariants by $B_{j_1,j_2}$. Below, we will first introduce divisors $Y_{j_1,j_2,r}$ of S. Afterwards (see Definition 6), we will define the divisors $D_{j_1,j_2}$ by means of the divisors $Y_{j_1,j_2,r}$.

For each $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ such that $j_2-j_1\ge 2$ and $1\le r\le \frac{j_2-j_1}{2}$, let $Y_{j_1,j_2,r}\subset S$ be the subvariety defined by the binomials of ${\mathcal {I}}_2(\text {Tor}(\Delta ^{(1)}))$ having the form

$$\begin{aligned} X_{i_1,j_1}X_{i_2,j_2} - X_{i'_1,j_1+r}X_{i'_2,j_2-r}. \end{aligned}$$

One can see that $Y_{j_1,j_2,r}$ is a $(\gamma -2)$-dimensional toric variety $\text {Tor}(\Omega _{j_1,j_2,r})$, where $\Omega _{j_1,j_2,r}\subset {\mathbb {R}}^{\gamma -2}$ is a full-dimensional lattice polytope (see Example 3 for a tangible instance). The (Euclidean) volume of this polytope equals

$$\begin{aligned} \frac{1}{(\gamma -2)!}(2(E_1+\cdots +E_{\gamma -1})-(E_{j_1}+E_{j_2}-\epsilon _{j_1,j_2,r})), \end{aligned}$$

where $\epsilon _{j_1,j_2,r}$ is defined as $\epsilon ^{(-)}_{j_1,j_2,r}+\epsilon ^{(+)}_{j_1,j_2,r}$, with

$$\begin{aligned} \epsilon ^{(-)}_{j_1,j_2,r}= & {} {\left\{ \begin{array}{ll} 0 \quad \text { if } &{} i^{(-)}(j_1+r)+i^{(-)}(j_2-r)\le i^{(-)}(j_1)+i^{(-)}(j_2) \\ 1 \quad \text { if } &{} i^{(-)}(j_1+r)+i^{(-)}(j_2-r) > i^{(-)}(j_1)+i^{(-)}(j_2) \\ \end{array}\right. } \\= & {} \max \left\{ 0, (i^{(-)}(j_1+r)+i^{(-)}(j_2-r)) - (i^{(-)}(j_1)+i^{(-)}(j_2)) \right\} , \end{aligned}$$

and

$$\begin{aligned} \epsilon ^{(+)}_{j_1,j_2,r}= & {} {\left\{ \begin{array}{ll} 0 \quad \text { if } &{} i^{(+)}(j_1+r)+i^{(+)}(j_2-r)\ge i^{(+)}(j_1)+i^{(+)}(j_2)\\ 1 \quad \text { if } &{} i^{(+)}(j_1+r)+i^{(+)}(j_2-r)< i^{(+)}(j_1)+i^{(+)}(j_2)\\ \end{array}\right. } \\= & {} \max \left\{ 0, (i^{(+)}(j_1)+i^{(+)}(j_2)) - (i^{(+)}(j_1+r)+i^{(+)}(j_2-r)) \right\} . \end{aligned}$$

In the above equalities for $\epsilon ^{(-)}_{j_1,j_2,r}$ and $\epsilon ^{(+)}_{j_1,j_2,r}$, we use the following result.

Lemma 2

The inequalities

$$\begin{aligned} i^{(-)}(j_1+r)+i^{(-)}(j_2-r)\le i^{(-)}(j_1)+i^{(-)}(j_2)+1 \end{aligned}$$

and

$$\begin{aligned} i^{(+)}(j_1+r)+i^{(+)}(j_2-r)\ge i^{(+)}(j_1)+i^{(+)}(j_2)-1 \end{aligned}$$

hold for all $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ such that $j_2-j_1\ge 2$ and $1\le r\le \frac{j_2-j_1}{2}$.

Proof

We only show the first inequality; the second one follows by symmetry. Consider the line segment $L=[(i^{(-)}(j_1),j_1),(i^{(-)}(j_2),j_2)]$, and let $(i',j_1+r)$ and $(i'',j_2-r)$ be the intersection points of L with the horizontal lines at heights $j_1+r$ and $j_2-r$. Note that L is contained in the interior of $\Delta$ and that $i'+i''=i^{(-)}(j_1)+i^{(-)}(j_2)$. If $i^{(-)}(j_1+r)+i^{(-)}(j_2-r)\ge i^{(-)}(j_1)+i^{(-)}(j_2)+2=i'+i''+2$, then $i'\le i^{(-)}(j_1+r)-1$ or $i''\le i^{(-)}(j_2-r)-1$, so $(i^{(-)}(j_1+r)-1,j_1+r)$ or $(i^{(-)}(j_2-r)-1,j_2-r)$ is a lattice point lying in the interior of $\Delta$. This is in contradiction with the definition of $i^{(-)}(\cdot )$. $\square$

Fig. 1

Picture of $\Delta$

Example 3

Assume that $\Delta = \Delta (f)$ is as in Fig. 1 (here $\gamma = 5$). Appropriate instances of $\Omega _{j_1,j_2,r}$ can be realized as in Fig. 2. Here, $\epsilon _{1,3,1}=1$ (since $\epsilon ^{(+)}_{1,3,1}=1$), $\epsilon _{1,4,1}=0$ and $\epsilon _{2,4,1}=1$ (since $\epsilon ^{(-)}_{2,4,1}=1$).

Fig. 2

Picture of the $\Omega _{j_1,j_2,r}$’s

The intersection of $Y_{j_1,j_2,r}$ with a typical fiber of $S\rightarrow {\mathbb {P}}^1$ is a quadratic hypersurface, hence there is a $B_{j_1,j_2,r}\in {\mathbb {Z}}$ such that $Y_{j_1,j_2,r}\sim 2H-B_{j_1,j_2,r}R$. Taking the intersection product of the latter equation with $H^{\gamma -2}$, we get

$$\begin{aligned} \deg Y_{j_1,j_2,r}= & {} Y_{j_1,j_2,r}\cdot H^{\gamma -2} = 2H^{\gamma -1}-B_{j_1,j_2,r}H^{\gamma -2}R \\= & {} 2(e_1+\ldots +e_{\gamma -1})-B_{j_1,j_2,r} \\= & {} 2(E_1+\ldots +E_{\gamma -1})-B_{j_1,j_2,r}, \end{aligned}$$

but $\deg Y_{j_1,j_2,r}=(\gamma -2)!\cdot \text {Vol}(\Omega _{j_1,j_2,r})$, so $B_{j_1,j_2,r}$ equals $E_{j_1}+E_{j_2}-\epsilon _{j_1,j_2,r}$.

Write

$$\begin{aligned} {\mathcal {S}}^{(-)}_{j_1,j_2}=\left\{ r\in \left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\} \,|\,\epsilon ^{(-)}_{j_1,j_2,r}=0\right\} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {S}}^{(+)}_{j_1,j_2}=\left\{ r\in \left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\} \,|\,\epsilon ^{(+)}_{j_1,j_2,r}=0\right\} . \end{aligned}$$

Definition 4

We say that $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$ if and only if there are no integers $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ with $j_2-j_1\ge 2$ such that ${\mathcal {S}}^{(-)}_{j_1,j_2}$ and ${\mathcal {S}}^{(+)}_{j_1,j_2}$ are non-empty and disjoint.

In other words, the condition ${\mathcal {P}}_1(v)$ means that for each pair of integers $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ with $j_2-j_1\ge 2$ either at least one of the sets ${\mathcal {S}}^{(-)}_{j_1,j_2}$, ${\mathcal {S}}^{(+)}_{j_1,j_2}$ is empty, or there is a common $r\in \left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\}$ for which $\epsilon ^{(-)}_{j_1,j_2,r}=\epsilon ^{(+)}_{j_1,j_2,r}=0$. There is a useful criterion to check whether ${\mathcal {S}}^{(-)}_{j_1,j_2}$ is empty or not (and analogously for ${\mathcal {S}}^{(+)}_{j_1,j_2}$): ${\mathcal {S}}^{(-)}_{j_1,j_2}=\emptyset$ if and only if all the lattice points $(i^{(-)}(j),j)$ with $j_1<j<j_2$ lie strictly right from the line segment $L=\left[ (i^{(-)}(j_1),j_1),(i^{(-)}(j_2),j_2)\right]$.

In the above definition, we also allow the lattice direction v to be different from (1, 0): in that case, first take a unimodular transformation $\chi$ such that $\chi (v)=(1,0)$ and that $\chi (\Delta )$ is contained in the horizontal strip $0\le Y\le \gamma$, and replace $\Delta$ by $\chi (\Delta )$ while checking the condition. The definition is independent of the particular choice of the unimodular transformation $\chi$.

In fact, in some of the examples below, the lattice direction v is (1, 0), but $\Delta$ is contained in a horizontal strip of the form $k\le Y\le k+\gamma$ with $k\ne 0$. In that case, we do not really need to apply any unimodular transformation $\chi$ first: we can define the sets ${\mathcal {S}}^{(-)}_{j_1,j_2}$ and ${\mathcal {S}}^{(+)}_{j_1,j_2}$ for $j_1,j_2\in \{k+1,\ldots ,k+\gamma -1\}$.

Fig. 3

Part of $\Delta ^{(1)}$

Example 5

Assume that a part of $\Delta ^{(1)}$ looks as in Fig. 3 (for some large enough n). In Table 1, the sets ${\mathcal {S}}^{(-)}_{j_1,j_2}$ and ${\mathcal {S}}^{(+)}_{j_1,j_2}$ are given for all couples $(j_1,j_2)$ with $j_1+j_2=15$ in this part of the polytope $\Delta ^{(1)}$. We conclude that $\Delta$ does not satisfy condition ${\mathcal {P}}_1(v)$ (consider $j_1=0$ and $j_2=15$).

Table 1 Table of subsets ${\mathcal {S}}^{(.)}_{j_1,j_2}$

For all polygons $\Delta$ that satisfy condition ${\mathcal {P}}_1(v)$, we give a recipe to construct the divisors $D_{j_1,j_2}$ in terms of the subvarieties $Y_{j_1,j_2,r}$.

Definition 6

Assume that the lattice polygon $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$.

• If ${\mathcal {S}}^{(-)}_{j_1,j_2}\cap {\mathcal {S}}^{(+)}_{j_1,j_2}\ne \emptyset$, we define $D_{j_1,j_2}$ as $Y_{j_1,j_2,r}$ with $r\in {\mathcal {S}}^{(-)}_{j_1,j_2}\cap {\mathcal {S}}^{(+)}_{j_1,j_2}$ minimal. Set $\epsilon _{j_1,j_2}=\epsilon _{j_1,j_2,r}=0$.

• If ${\mathcal {S}}^{(-)}_{j_1,j_2}=\emptyset$ and ${\mathcal {S}}^{(+)}_{j_1,j_2}\ne \emptyset$ or vice versa, take $r\in {\mathcal {S}}^{(-)}_{j_1,j_2}\cup {\mathcal {S}}^{(+)}_{j_1,j_2}$ minimal, define $D_{j_1,j_2}=Y_{j_1,j_2,r}$ and set $\epsilon _{j_1,j_2}=\epsilon _{j_1,j_2,r}=1$.

• If ${\mathcal {S}}^{(-)}_{j_1,j_2}={\mathcal {S}}^{(+)}_{j_1,j_2}=\emptyset$, define $D_{j_1,j_2}=Y_{j_1,j_2,1}$ and set $\epsilon _{j_1,j_2}=\epsilon _{j_1,j_2,1}=2$.

Remark 7

In Definition 6, the divisor $D_{j_1,j_2}$ is always of the form $Y_{j_1,j_2,r}$ and r is chosen such that $\epsilon _{j_1,j_2,r}$ is minimal, or equivalently, $B_{j_1,j_2,r}$ is maximal. Moreover, if $D_{j_1,j_2}=Y_{j_1,j_2,r}$ and if we define $\epsilon _{j_1,j_2}^{(-)}=\epsilon _{j_1,j_2,r}^{(-)}$ and $\epsilon _{j_1,j_2}^{(+)}=\epsilon _{j_1,j_2,r}^{(+)}$, then

$$\begin{aligned} \epsilon _{j_1,j_2}^{(-)} = \min _s \epsilon _{j_1,j_2,s}^{(-)}, \quad \epsilon _{j_1,j_2}^{(+)}=\min _t \epsilon _{j_1,j_2,t}^{(+)} \quad \text { and } \quad \epsilon _{j_1,j_2} = \epsilon _{j_1,j_2}^{(-)} + \epsilon _{j_1,j_2}^{(+)}. \end{aligned}$$

(6)

Here, it is crucial that $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$: if ${\mathcal {S}}^{(-)}_{j_1,j_2}$ and ${\mathcal {S}}^{(+)}_{j_1,j_2}$ were non-empty and disjoint, then $\min _r \epsilon _{j_1,j_2,r} = 1$ (take $r\in {\mathcal {S}}^{(-)}_{j_1,j_2}\cup {\mathcal {S}}^{(+)}_{j_1,j_2}$), but $\min _s \epsilon _{j_1,j_2,s}^{(-)}=\min _t \epsilon _{j_1,j_2,t}^{(+)}=0$.

If we set $B_{j_1,j_2}=E_{j_1}+E_{j_2}-\epsilon _{j_1,j_2}$, we have that $D_{j_1,j_2}\sim 2H-B_{j_1,j_2}R$ and

$$\begin{aligned} {\sum }_{j_2-j_1\ge 2}\,B_{j_1,j_2}= & {} (\gamma -4)(E_1+\cdots +E_{\gamma -1})+E_1+E_{\gamma -1}- {\sum }_{j_2-j_1\ge 2}\,\epsilon _{j_1,j_2}\\= & {} (\gamma -4)(g-\gamma +1)+E_1+E_{\gamma -1}-{\sum }_{j_2-j_1\ge 2}\,\epsilon _{j_1,j_2}. \end{aligned}$$

Example 8

If $\partial \Delta ^{(1)}$ meets each horizontal line of height $j\in \{2,\ldots ,\gamma -2\}$ in two lattice points, we have $\epsilon _{j_1,j_2,r}=0$ and ${\mathcal {S}}^{(-)}_{j_1,j_2}={\mathcal {S}}^{(+)}_{j_1,j_2}=\left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\}$ for all $j_1,j_2,r$. Hence, $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$. Moreover, $\epsilon _{j_1,j_2}=0$ and $D_{j_1,j_2}=Y_{j_1,j_2,1}$ for all $j_1,j_2$. In this case,

$$\begin{aligned} \sharp (\partial \Delta ^{(1)}\cap {\mathbb {Z}}^2)=(E_1+1)+(E_{\gamma -1}+1)+2(\gamma -3) \end{aligned}$$

and $\sum \epsilon _{j_1,j_2}=0$, so

$$\begin{aligned} \sum B_{j_1,j_2}=(\gamma -4)g-(\gamma ^2-3\gamma )+\sharp (\partial \Delta ^{(1)}\cap {\mathbb {Z}}^2). \end{aligned}$$

Remark 9

If S is singular, let $\lambda >0$ be an integer and consider the inclusions from (5). Note that $\Delta [\lambda ]$ satisfies condition ${\mathcal {P}}_1(v)$ if and only if $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$. We can define the subvarieties $Y_{j_1,j_2,r}$ and $D_{j_1,j_2}$ of $S'$ in the same way as we did before (using $\Delta [\lambda ]$ instead of $\Delta$). Since $H'=H+\lambda R$, we get that

$$\begin{aligned} Y_{j_1,j_2,r}\sim 2H'-((E_{j_1}+\lambda )+(E_{j_2}+\lambda )-\epsilon _{j_1,j_2,r})R=2H-B_{j_1,j_2,r}R \end{aligned}$$

and $D_{j_1,j_2}\sim 2H-B_{j_1,j_2}R$.

We are now able to state and prove the main result of this subsection.

Theorem 10

If $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$, there exist ${\gamma -2 \atopwithdelims ()2}$ effective divisors $D_{j_1,j_2}$ on ${\mathbb {P}}({\mathcal {E}})$ (with $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ and $j_2-j_1\ge 2$) such that

• $T'$ is the (scheme-theoretical) intersection of the divisors $D_{j_1,j_2}$,

• $D_{j_1,j_2}\sim 2H-B_{j_1,j_2}R$ for all $j_1,j_2$, where $B_{j_1,j_2} = E_{j_1} + E_{j_2} - \epsilon _{j_1,j_2}$, and,

• $\sum _{j_2-j_1\ge 2}\,B_{j_1,j_2} = (\gamma -4)g-(\gamma ^2-3\gamma ) +\sharp (\partial \Delta ^{(1)}\cap {\mathbb {Z}}^2)$.

Proof

By Remark 9, we may assume that S is smooth, hence ${\mathbb {P}}({\mathcal {E}})\cong S$. We need to prove that ${\mathcal {I}}(\text {Tor}(\Delta ^{(1)}))={\mathcal {I}}(\bigcap D_{j_1,j_2})$, where the inclusion ${\mathcal {I}}(\bigcap D_{j_1,j_2})\subset {\mathcal {I}}(\text {Tor}(\Delta ^{(1)}))$ is trivial. Pick an arbitrary quadratic binomial

$$\begin{aligned} f=X_{i_1,j_1}X_{i_2,j_2}-X_{i_3,j_3}X_{i_4,j_4}\in {\mathcal {I}}(\text {Tor}(\Delta ^{(1)})). \end{aligned}$$

These binomials generate the ideal, so we only need to show that $f\in {\mathcal {I}}(\bigcap D_{j_1,j_2})$. Note that $j_1+j_2=j_3+j_4$, so we may assume that $j_1\le j_3\le j_4\le j_2$. Moreover, if $j_1=j_3$ and $j_4=j_2$, we get that $f\in {\mathcal {I}}(S)\subset {\mathcal {I}}(\bigcap D_{j_1,j_2})$. So we may even assume that $j_1<j_3$.

Take r such that $D_{j_1,j_2}=Y_{j_1,j_2,r}$. We claim that

$$\begin{aligned} I:=i_1+i_2=i_3+i_4\ge i^{(-)}(j_1+r)+i^{(-)}(j_2-r). \end{aligned}$$

If $\epsilon _{j_1,j_2}^{(-)}=\epsilon _{j_1,j_2,r}^{(-)}=0$, this follows from

$$\begin{aligned} I\ge i^{(-)}(j_1)+i^{(-)}(j_2)\ge i^{(-)}(j_1+r)+i^{(-)}(j_2-r). \end{aligned}$$

If $\epsilon _{j_1,j_2}^{(-)}=\epsilon _{j_1,j_2,r}^{(-)}=1$, we have that $\epsilon _{j_1,j_2,j_3-j_1}^{(-)}=1$ by (6) (since $\Delta$ satisfies condition ${\mathcal {P}}_1(v)$), hence

$$\begin{aligned} I\ge i^{(-)}(j_3)+i^{(-)}(j_4)=i^{(-)}(j_1+r)+i^{(-)}(j_2-r), \end{aligned}$$

where we use Lemma 2. Analogously, we can show that $I\le i^{(+)}(j_1+r)+i^{(+)}(j_2-r)$.

The above claim implies that we can find integers $i'_1,i'_2$ such that $i'_1+i'_2=I$, $i^{(-)}(j_1+r)\le i'_1 \le i^{(+)}(j_1+r)$, $i^{(-)}(j_2-r)\le i'_2 \le i^{(+)}(j_2-r)$, hence

$$\begin{aligned} X_{i_1,j_1}X_{i_2,j_2}-X_{i'_1,j_1+r}X_{i'_2,j_2-r}\in {\mathcal {I}}(D_{j_1,j_2})={\mathcal {I}}(Y_{j_1,j_2,r}). \end{aligned}$$

So we may replace the term $X_{i_1,j_1}X_{i_2,j_2}$ in f by $X_{i'_1,j_1+r}X_{i'_2,j_2-r}$ (and in particular, $j_1$ by $j_1+r$ and $j_2$ by $j_2-r$). Continuing in this way, we will eventually get that $j_1=j_3$ and $j_4=j_2$, hence $f\in {\mathcal {I}}(S)$. This will happen after a finite number of steps since the maximum of $j_2-j_1$ and $j_4-j_3$ decreases after each step.

We are left with proving the formula for the sum of the $B_{j_1,j_2}$’s. By Remark 7 and the elaboration of Example 8, it suffices to show that the sum of the $\epsilon _{j_1,j_2}$ counts the number of times that $\partial \Delta ^{(1)}$ intersects the horizontal lines of height $2,\ldots ,\gamma -2$ in a non-lattice point. Let $A^{(-)}$ be the set of couples $(j_1,j_2)$ such that $j_1,j_2\in \{1,\ldots ,\gamma -1\}$, $j_2-j_1\ge 2$ and ${\mathcal {S}}_{j_1,j_2}^{(-)}=\emptyset$ (or equivalently, the line segment $L=\left[ (i^{(-)}(j_1),j_1),(i^{(-)}(j_2),j_2)\right]$ passes left from all the lattice points $(i^{(-)}(j'),j')$ with $j_1<j'<j_2$). Let $B^{(-)}$ be the set of integers $j\in \{1,\ldots ,\gamma -1\}$ such that $(i^{(-)}(j),j)\not \in \partial \Delta ^{(1)}$. We claim that the sets $A^{(-)}$ and $B^{(-)}$ have the same cardinality. We will do this by giving a concrete bijection between these sets. Analogously, we can define the sets $A^{(+)}$ and $B^{(+)}$, and prove that they have the same number of elements. The theorem follows directly, since $\sharp (A^{(-)}\cup A^{(+)})=\sum _{j_1,j_2}\,\epsilon _{j_1,j_2}$ by (6) and $\sharp (B^{(-)}\cup B^{(+)})$ is the number of non-lattice point intersections.

Fig. 4

The line segments L, $L'$ and $L''$

If $(j_1,j_2)\in A^{(-)}$, then the line segment $L=[(i^{(-)}(j_1),j_1),(i^{(-)}(j_2),j_2)]$ will pass at the left hand side of the lattice points $(i^{(-)}(j),j)$ with $j_1<j<j_2$. For precisely one of these lattice points, the horizontal distance to L will be equal to the minimal value $\frac{1}{j_2-j_1}$. Consider the map

$$\begin{aligned} \alpha ^{(-)}:A^{(-)}\rightarrow B^{(-)} \end{aligned}$$

sending the couple $(j_1,j_2)$ to the value of j of that lattice point; see below for an example. On the other hand, if $j\in B^{(-)}$, thus $(i^{(-)}(j),j)\not \in \partial \Delta ^{(1)}$, then there should be lattice points $(i^{(-)}(j_1),j_1)$ and $(i^{(-)}(j_2),j_2)$ with $j_1<j<j_2$ such that $L=[(i^{(-)}(j_1),j_1),(i^{(-)}(j_2),j_2)]$ passes left from $(i^{(-)}(j),j)$. If we take a couple $(j_1,j_2)$ that satisfies this property and has a minimal value for $j_2-j_1$, then $(j_1,j_2)\in A^{(-)}$. Indeed, if $(i^{(-)}(j'),j')$ with $j_1<j'<j_2$ lies on or left from L, then either $(j_1,j')$ or $(j',j_2)$ would also satisfy the condition and would have a smaller value for the difference of the heights. Now let’s show that the couple $(j_1,j_2)$ is unique. If not, there exists another couple $(j_1',j_2')\in A^{(-)}$ with $j_1'<j<j_2'$ such that $L'=[(i^{(-)}(j_1'),j_1'),(i^{(-)}(j_2'),j_2')]$ passes left from $(i^{(-)}(j),j)$ with $j_2'-j_1'=j_2-j_1$. We may assume that $j_1'<j_1<j<j_2'<j_2$. Then L passes left from $(i^{(-)}(j_2'),j_2')$ and $L'$ passes left from $(i^{(-)}(j_1),j_1)$, so $L''=[(i^{(-)}(j_1'),j_1'),(i^{(-)}(j_2),j_2)]$ passes left from all the lattice points $(i^{(-)}(j'),j')$ with $j_1'<j'<j_2$ (see Fig. 4). Let’s denote the horizontal distance from the line segment $L''$ to the lattice point $(i^{(-)}(j'),j')$ by $d(j')$. Using Lemma 2, we obtain that $d(j_1'+r)+d(j_2-r)=1$ for all $1\le r\le \frac{j_2-j_1'}{2}$. For precisely one integer $j_1'<j'<j_2$, the distance $d(j')$ is equal to the minimal value $\frac{1}{j_2-j_1'}$. On the other hand, except for $j'=j_1$ or $j'=j_2'$, the distance $d(j')$ has to be at least $\frac{1}{j_2-j_1}=\frac{1}{j_2'-j_1'}$, since $(i^{(-)}(j'),j')$ lies strictly right from L or $L'$. So we may assume that $j'=j_1$ (the case $j'=j_2'$ is analogous), hence $d(j_1)=\frac{1}{j_2-j_1'}$ and $d(j_2')=1-\frac{1}{j_2-j_1'}$ (using $r=j_1-j_1'$ above). It follows that the horizontal distance to $L''$ from the point on $L'$ on height $j_1$ is equal to

$$\begin{aligned} \frac{j_1-j_1'}{j_2'-j_1'}\cdot d(j_2')=\frac{j_1-j_1'}{j_2'-j_1'}\cdot \frac{j_2-j_1'-1}{j_2-j_1'}\ge \frac{j_1-j_1'}{j_2-j_1'} \ge \frac{1}{j_2-j_1'}=d(j_1), \end{aligned}$$

so $L'$ does not pass left from $(i^{(-)}(j_1),j_1)$, a contradiction. In conclusion we can consider the map

$$\begin{aligned} \beta ^{(-)}:B^{(-)}\rightarrow A^{(-)} \end{aligned}$$

sending j to the unique such couple $(j_1,j_2)$.

The maps $\alpha ^{(-)}$ and $\beta ^{(-)}$ are inverse of each other. For instance, to prove that the map $\alpha ^{(-)}\circ \beta ^{(-)}$ is the identity map on $B^{(-)}$, consider $j\in B^{(-)}$ and write $\beta ^{(-)}(j)=(j_1,j_2)$. If $\alpha ^{(-)}(j_1,j_2)=j'\ne j$, then the horizontal distance from $(i^{(-)}(j),j)$ to $L=[(i^{(-)}(j_1),j_1),(i^{(-)}(j_2),j_2)]$ is of the form $\frac{d}{j_2-j_1}$ with $1<d<j_2-j_1$. But then either $L'=[(i^{(-)}(j'),j'),(i^{(-)}(j_2),j_2)]$ (the case $j'<j$) or $L''=[(i^{(-)}(j_1),j_1),(i^{(-)}(j'),j')]$ (the case $j'>j$) passes left from $(i^{(-)}(j),j)$. This is in contradiction with $\beta ^{(-)}(j)=(j_1,j_2)$, since $j_2-j'$ and $j'-j_1$ are both strictly smaller than $j_2-j_1$. We leave the proof of the equality $\beta ^{(-)}\circ \alpha ^{(-)}=\text {Id}_{A^{(-)}}$ as an exercise. $\square$

Fig. 5

Part of $\partial \Delta ^{(1)}$

Example 11

Consider a polygon $\Delta$ of which a part of the boundary of $\Delta ^{(1)}$ is as in Fig. 5 (the (i, j)-coordinates are translated a bit). For this horizontal slice of the polygon,

$$\begin{aligned} A^{(-)}=\{(0,2),(0,7),(2,4),(2,7),(4,6),(4,7)\} \quad \text { and } \quad B^{(-)}=\{1,2,3,4,5,6\}. \end{aligned}$$

The map $\alpha ^{(-)}$ is defined as follows:

$$\begin{aligned} (0,2)\mapsto 1\,\ (0,7)\mapsto 2\,\ (2,4)\mapsto 3\,\ (2,7)\mapsto 4\,\ (4,6)\mapsto 5\,\ (4,7)\mapsto 6. \end{aligned}$$

One can show that the $D_{j_1,j_2}$’s in Theorem 10 can be used to resolve ${\mathcal {O}}_{T'}$ as an ${\mathcal {O}}_{{\mathbb {P}}({\mathcal {E}})}$-module, following Schreyer [9]. For this one needs that the fibers of $\pi |_{T'}:T'\rightarrow {\mathbb {P}}^1$ have constant Betti numbers and that the corresponding resolutions are pure, but this can be verified. So it is justified to call the $B_{j_1,j_2}$’s the first scrollar Betti numbers of the toric surface $\text {Tor}(\Delta ^{(1)})$, even though we will not push this discussion further.

2.4 First scrollar Betti numbers of non-degenerate curves relative to the toric surface

We will use the same set-up and assumptions as in the beginning of Sect. 2.3. The assumption (i) implies that $\Delta ^{(2)}\ne \emptyset$. Moreover, we also use the notations appearing in the inclusions (5), so $C'$ and $T'$ are the strict transforms of the canonically embedded $\Delta$-non-degenerate curve C and the toric surface T under the resolution $\mu :S'\cong {\mathbb {P}}({\mathcal {E}}) \rightarrow S$. In this section, we will present divisors on $S'$ that scheme-theoretically cut out $C'$ from $T'$.

Herefore, we rely on the following construction from [4]. Write

$$\begin{aligned} f = \sum _{(i,j) \in \Delta \cap {\mathbb {Z}}^2} c_{i,j} x^iy^j \in k[x^{\pm 1},y^{\pm 1}] \end{aligned}$$

and consider $w\in \Delta ^{(2)}\cap {\mathbb {Z}}^2$. For each $(i,j)\in \Delta \cap {\mathbb {Z}}^2$, there exist $u_{i,j},v_{i,j}\in \Delta ^{(1)}\cap {\mathbb {Z}}^2$ such that $(i,j) - w = (u_{i,j} - w) + (v_{i,j} - w ).$ Hereby, we use that $\Delta +\Delta ^{(2)}\subset 2\Delta ^{(1)}$ and that the polygon $\Delta ^{(1)}$ is normal. Then the quadrics

$$\begin{aligned} Q_w=\sum _{(i,j) \in \Delta \cap {\mathbb {Z}}^2} c_{i,j} X_{u_{i,j}} X_{v_{i,j}} \in k[X_{i,j}]_{(i,j)\in \Delta '^{(1)}\cap {\mathbb {Z}}^2}, \end{aligned}$$

where w ranges over $\Delta ^{(2)}\cap {\mathbb {Z}}^2$, scheme-theoretically cut out C from T.

In order to create the divisors $D_\ell$, we will need an extra condition on $\Delta$, which garantees that we can choose the lattice points $u_{i,j},v_{i,j}$ in a particular way.

Definition 12

We say that $\Delta$ satisfies condition ${\mathcal {P}}_2(v)$ if for each lattice point (i, j) of $\Delta$ and each horizontal line L, there exist two (not necessarily distinct) horizontal lines $M_1,M_2$, such that for all $w \in L \cap \Delta ^{(2)} \cap {\mathbb {Z}}^2$, there exist $u_{i,j} \in M_1 \cap \Delta ^{(1)} \cap {\mathbb {Z}}^2$ and $v_{i,j} \in M_2 \cap \Delta ^{(1)} \cap {\mathbb {Z}}^2$ (dependent on (i, j) and w) such that

$$\begin{aligned} (i,j) - w = (u_{i,j} - w) + (v_{i,j} - w ). \end{aligned}$$

Remark 13

Write

$$\begin{aligned} i^{(--)}(j)=\min \{i\in {\mathbb {Z}}\,|\,(i,j)\in \Delta ^{(2)}\}\quad \text { and }\quad i^{(++)}(j)=\max \{i\in {\mathbb {Z}}\,|\,(i,j)\in \Delta ^{(2)}\} \end{aligned}$$

for all $j\in \{2,\ldots ,\gamma -2\}$. An equivalent definition is as follows: $\Delta$ satisfies condition ${\mathcal {P}}_2(v)$ if and only if for all $(i,j)\in \Delta$ and for all $j'\in \{2,\ldots ,\gamma -2\}$, there exist $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ such that $j_1+j_2=j+j'$ and

$$\begin{aligned} i+\left[ i^{(--)}(j'),i^{(++)}(j')\right] \subset \left[ i^{(-)}(j_1), i^{(+)}(j_1)\right] + \left[ i^{(-)}(j_2),i^{(+)}(j_2)\right] . \end{aligned}$$

(7)

This condition is obviously satisfied for $(i,j) \in \Delta ^{(1)}$ (take $j_1=j$ and $j_2=j'$). Moreover, the condition also holds if (i, j) lies on the interior of a horizontal edge (i.e. the top or bottom edge) of $\Delta$. Indeed, assume for instance that (i, j) lies in the interior of the top edge $[(i^-,j),(i^+,j)]$ of $\Delta$. We have that

$$\begin{aligned} i^{(-)}(j'+1)+i^{(-)}(j-1)\le i^{(--)}(j')+i^-+1\le i^{(--)}(j')+i. \end{aligned}$$

Hereby, the first inequality follows by replacing L in the proof of Lemma 2 by the half-closed line segment $[(i^{(--)}(j'),j'),(i^-,j)[$. Analogously, we get that

$$\begin{aligned} i^{(+)}(j'+1)+i^{(+)}(j-1)\ge i^{(++)}(j')+i, \end{aligned}$$

so (7) follows for $j_1=j'+1$ and $j_2=j-1$.

Although at first sight the condition ${\mathcal {P}}_2(v)$ might seem strong, it is not so easy to cook up instances of lattice polygons $\Delta$ for which the condition is not satisfied. The smallest example we have found is a polygon with 46 interior lattice points and lattice width 10.

Fig. 6

A lattice polygon $\Delta$ that does not satisfy condition ${\mathcal {P}}_2(v)$

Example 14

Let $\Delta$ be as in Fig. 6 (the dashed line indicates $\Delta ^{(1)}$). We claim that $\Delta$ does not satisfy condition ${\mathcal {P}}_2(v)$. Indeed, take the top vertex $(i,j)=(4,10)$ of $\Delta$ and the horizontal line L at height 6. For the point $w\in L \cap \Delta ^{(2)} \cap {\mathbb {Z}}^2$, consider the bold-marked lattice points (3, 6) and (6, 6) on L. In both cases, there is a unique decomposition of $(i,j) - w$:

$$\begin{aligned} (1,4) = (0,1) + (1,3) \qquad \text {resp.} \qquad (-2,4) = (-1,2) + (-1,2). \end{aligned}$$

So one sees that it is impossible to take the $u_{i,j}$’s and/or the $v_{i,j}$’s on the same line, which proves the claim.

Theorem 15

If $\Delta$ satisfies condition ${\mathcal {P}}_2(v)$, then there exist $\gamma -3$ effective divisors $D_\ell$ on ${\mathbb {P}}({\mathcal {E}})$ (with $2\le \ell \le \gamma -2$) such that

• $C'$ is the (scheme-theoretical) intersection of $T'$ and the divisors $D_\ell$,

• $D_\ell \sim 2H-B_\ell R$ for all $\ell$, where

  $$\begin{aligned} B_\ell =i^{(++)}(\ell )-i^{(--)}(\ell )=-1+\sharp \{(i,j)\in \Delta ^{(2)}\cap {\mathbb {Z}}^2\,|\,j=\ell \}, \end{aligned}$$

  so

  $$\begin{aligned} \sum _{2\le \ell \le \gamma -2}\,B_\ell =\sharp (\Delta ^{(2)}\cap {\mathbb {Z}}^2)-(\gamma -3). \end{aligned}$$

Proof

The formula for the sum $\sum _\ell \,B_\ell$ is easily verified, so we focus on the other assertions. Take $\lambda \ge 0$ so that $S'=S(e_1+\lambda ,\ldots ,e_{\gamma -1}+\lambda )$ is smooth (and isomorphic to ${\mathbb {P}}({\mathcal {E}})$) and define $\Delta '=\Delta [\lambda ]$. We are going to use the inclusions

$$\begin{aligned} C'\subset T'\subset S'\subset {\mathbb {P}}^{g-1+\lambda (\gamma -1)} \end{aligned}$$

from (5), where $T'=\text {Tor}(\Delta '^{(1)})$. Write $(X_{i,j})_{(i,j)\in \Delta '^{(1)}\cap {\mathbb {Z}}^2}$ for the projective coordinates on ${\mathbb {P}}^{g-1+\lambda (\gamma -1)}$.

Let $\ell \in \{2,\ldots ,\gamma -2\}$ and denote the lattice points of $\Delta [2\lambda ]^{(2)}$ of height $\ell$ by $w_0,\ldots ,w_{B_\ell +2\lambda }$. If $w\in \{w_0,\ldots ,w_{B_\ell +2\lambda }\}$ and $(i,j)\in \Delta$, then we claim that we can find $u_{i,j}, v_{i,j}\in \Delta '^{(1)}$ such that $(i,j)-w=(u_{i,j}-w)+(v_{i,j}-w)$, in such a way that their second coordinates are independent from w. Indeed, since $\Delta$ satisfies condition ${\mathcal {P}}_2(v)$, there exist $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ such that $j_1+j_2=j+\ell$ and

$$\begin{aligned} i+[i^{(--)}(\ell ),i^{(++)}(\ell )]\subset \left[ i^{(-)}(j_1), i^{(+)}(j_1)\right] + \left[ i^{(-)}(j_2),i^{(+)}(j_2)\right] , \end{aligned}$$

hence

$$\begin{aligned} i+[i^{(--)}(\ell ),i^{(++)}(\ell )+2\lambda ]\subset \left[ i^{(-)}(j_1), i^{(+)}(j_1)+\lambda \right] + \left[ i^{(-)}(j_2),i^{(+)}(j_2)+\lambda \right] . \end{aligned}$$

This implies that we can take $u_{i,j}$ and $v_{i,j}$ with second coordinates $j_1$ and $j_2$. Define

$$\begin{aligned} Q_w=\sum _{(i,j) \in \Delta \cap {\mathbb {Z}}^2} c_{i,j} X_{u_{i,j}} X_{v_{i,j}} \in k[X_{i,j}]_{(i,j)\in \Delta '^{(1)}\cap {\mathbb {Z}}^2}. \end{aligned}$$

A consequence of the choice of $u_{i,j}, v_{i,j}$ is that

$$\begin{aligned} X_{w_{s}}Q_{w_{r+1}}-X_{w_{s+1}}Q_{w_r} \in {\mathcal {I}}(S') \end{aligned}$$

(rather than just ${\mathcal {I}}(T')$) for all $r\in \{0,\ldots ,B_\ell +2\lambda -1\}$ and $s\in \{0,\ldots ,B_\ell +\lambda -1\}$. Since

$$\begin{aligned} \frac{X_{w_1}}{X_{w_0}} = \frac{X_{w_2}}{X_{w_1}} = \cdots = \frac{X_{w_{B_\ell +\lambda }}}{X_{w_{B_\ell +\lambda - 1}}} \end{aligned}$$

is a local parameter for the $(\gamma -2)$-plane $R_{(0:1)} = \pi ^{-1}(0:1) \subset S'$, it follows that the $R_{(0:1)}$-orders of

$$\begin{aligned} {\mathcal {Z}}(Q_{w_1}), {\mathcal {Z}}(Q_{w_2}), \dots , {\mathcal {Z}}(Q_{w_{B_\ell +2\lambda }}) \end{aligned}$$

(8)

increase by 1 at each step. For a similar reason, with $R_{(1:0)} = \pi ^{-1}(1:0)\subset S'$, the $R_{(1:0)}$-orders of (8) decrease by 1 at each step. We conclude that there exists an effective divisor $D_\ell$ such that for all $i\in \{0,\dots ,B_\ell +2\lambda \}$ we have

$$\begin{aligned} {\mathcal {Z}}(Q_{w_i}) \, = \, i \cdot R_{(0:1)} \, + \, (B_\ell + 2\lambda - i) \cdot R_{(1:0)} \, + \, D_{\ell } \end{aligned}$$

(9)

on $S'$. The divisor $D_\ell$ is in fact the divisor of $S'$ cut out by the quadrics in (8). Using (9) and Remark 9, we get that

$$\begin{aligned} D_\ell \sim 2H' - (B_\ell +2\lambda ) R=2H-B_\ell R, \end{aligned}$$

so it is sufficient to show that the quadrics $Q_w$ (where w ranges over $\Delta [2\lambda ]^{(2)}\cap {\mathbb {Z}}^2$) cut out $C'$ from $T'$. If $\lambda =0$, this follows from [4, Theorem 3.3].

Before we prove this, we need to introduce one more notion: for each lattice polygon $\Gamma$ with two-dimensional $\Gamma ^{(1)}$, write $\Gamma ^{max}$ to denote the largest lattice polygon with interior lattice polygon equal to $\Gamma ^{(1)}$, so $\Gamma ^{max}\supset \Gamma$. The polygon $\Gamma ^{max}$ can be constructed as follows. Let $v_1,\ldots ,v_r$ be the primitive inward pointing normal vectors of the edges of $\Gamma ^{(1)}$ and write $\Gamma ^{(1)}$ as an intersection $\cap _{t=1}^r\,H_t$ of half-planes

$$\begin{aligned} H_t=\{P\in {\mathbb {R}}^2\,|\,\langle P,v_t\rangle \ge a_t\} \end{aligned}$$

(where $\langle \cdot ,\cdot \rangle$ denotes the standard inner product and $a_t\in {\mathbb {Z}}$). Then

$$\begin{aligned} \Gamma ^{max}=\cap _{t=1}^r\,H_t^{(-1)} \quad \text {with} \quad H_t^{(-1)}=\left\{ P\in {\mathbb {R}}^2\,|\,\langle P,v_t\rangle \ge a_t-1\right\} . \end{aligned}$$

We will use the following two properties (see [3, Section 2] for other properties of $\Gamma ^{max}$):

• If $\Gamma ^{(2)}\ne \emptyset$, then $2\Gamma ^{(1)}=\Gamma ^{(2)}+\Gamma ^{max}$, since both lattice polygons are defined by the half-planes $2H_t=\{P\in {\mathbb {R}}^2\,|\,\langle P,v_t\rangle \ge -2a_t\}$.

• If $\Phi _1,\Phi _2$ are lattice polygons such that $\Phi _1 + \Gamma \subset \Phi _2 + \Gamma ^{max}$, then $\Phi _1\subset \Phi _2$ if $\Phi _2$ satisfies the following condition: it is the intersection of half-planes $H'_t$ with $H'_t$ of the form

  $$\begin{aligned} \left\{ P\in {\mathbb {R}}^2\,|\,\langle P,v_t\rangle \ge b_t\right\} \end{aligned}$$

  for some $b_t\in {\mathbb {Z}}$ (hence parallel to $H_t$).

  Indeed, if $\Phi _1\not \subset \Phi _2$, take a lattice point P in $\Phi _1\setminus \Phi _2$. Then $\langle P,v_t\rangle <b_t$ for some value of t. Take $Q\in \Gamma$ with $\langle Q,v_t\rangle =a_t-1$ (this is always possible). We have that $P+Q\in \Phi _1+\Gamma$ and $\langle P+Q,v_t\rangle <a_t+b_t-1$, but $\Phi _2 + \Gamma ^{max}$ is the intersection of the half-planes $H''_t=\{P\in {\mathbb {R}}^2\,|\,\langle P,v_t\rangle \ge a_t+b_t-1\}$, so $P+Q\not \in \Phi _2 + \Gamma ^{max}$, a contradiction.

Now take $F\in {\mathcal {I}}(C')$ homogeneous of degree d and let

$$\begin{aligned} \xi : k[X_{i,j}]_{(i,j)\in \Delta '^{(1)}\cap {\mathbb {Z}}^2} \rightarrow k[x^{\pm 1},y^{\pm 1}] \end{aligned}$$

be the ring morphism that maps $X_{i,j}$ to $x^i y^j$. Since $\xi (F)(x,y)=0$ for all $(x,y)\in {\mathbb {T}}^2$ with $f(x,y)=0$ (and f is irreducible), the Laurent polynomial $\xi (F)$ has to be of the form cf for some $c\in k[x^{\pm 1},y^{\pm 1}]$. The Newton polygon of cf is equal to $\Delta (c)+\Delta$, while the Newton polygon $\Delta (\xi (F))$ is contained in

$$\begin{aligned} d\Delta '^{(1)}=(d-2)\Delta '^{(1)}+\Delta '^{(2)}+\Delta '^{max}=(d-2)\Delta '^{(1)}+\Delta [2\lambda ]^{(2)}+\Delta ^{max} \end{aligned}$$

(here, we use the first property of maximal polygons with $\Gamma =\Delta '$). So we obtain that

$$\begin{aligned} \Delta (c)+\Delta \subset (d-2)\Delta '^{(1)}+\Delta [2\lambda ]^{(2)}+\Delta ^{max}. \end{aligned}$$

Now we can use the second property of maximal polygons with $\Phi _1=\Delta (c)$, $\Phi _2=(d-2)\Delta '^{(1)}+\Delta [2\lambda ]^{(2)}$ and $\Gamma =\Delta$. Note that $\Phi _2$ might have a horizontal (top or bottom) edge while $\Delta ^{(1)}$ has not, but this is not an issue (since $\Delta ^{(1)}\not \cong k\Sigma$). It follows that

$$\begin{aligned} \Delta (c)\subset (d-2)\Delta '^{(1)}+\Delta [2\lambda ]^{(2)}. \end{aligned}$$

So we can write

$$\begin{aligned} c=\sum _{w=(i,j)\in \Delta [2\lambda ]^{(2)}\cap {\mathbb {Z}}^2}\, g_{i,j}x^i y^j \end{aligned}$$

for polynomials $g_{i,j}\in k[x,y]$ with $\Delta (g_{i,j})\subset (d-2)\Delta '^{(1)}$. For all lattice points $w=(i,j)\in \Delta [2\lambda ]^{(2)}\cap {\mathbb {Z}}^2$, there is a homogeneous polynomial $G_{i,j}\in k[X_{i,j}]_{(i,j)\in \Delta '^{(1)}\cap {\mathbb {Z}}^2}$ such that $\xi (G_{i,j})=g_{i,j}$. On the other hand, $\xi (Q_w)=x^i y^j f$, hence

$$\begin{aligned} \xi (F)=cf=\sum _{w=(i,j)\in \Delta [2\lambda ]^{(2)}\cap {\mathbb {Z}}^2}\, \xi (G_{i,j})\xi (Q_w). \end{aligned}$$

So $F-\sum _{w=(i,j)}\, G_{i,j}Q_w$ belongs to the kernel of the map $\xi$, which implies that it is contained in ${\mathcal {I}}_d(T')$, which is what we wanted to prove. $\square$

2.5 First scrollar Betti numbers for non-degenerate curves

We are ready to prove the main result of this section, by combining the results from Sects. 2.3 and 2.4.

Theorem 16

Let $\Delta$ be a lattice polygon with $\text {lw}(\Delta )\ge 4$ such that $\Delta ^{(1)}\not \cong \Upsilon$ and $\Delta ^{(1)}\not \cong k\Sigma$ for any integer k. Assume that $\Delta$ satisfies the conditions ${\mathcal {P}}_1(v)$ and ${\mathcal {P}}_2(v)$, where v is a lattice-width direction. Let C be a $\Delta$-non-degenerate curve and let $g_\gamma ^1$ be the combinatorial gonality pencil on C corresponding to v (with $\gamma =\text {lw}(\Delta )$). Then the first scrollar Betti numbers of C with respect to $g_\gamma ^1$ are given by

$$\begin{aligned} \{B_\ell \}_{\ell \in \{2,\ldots ,\gamma -2\}}\cup {\{B_{j_1,j_2}\}}_{\begin{array}{c} j_1,j_2\in \{1,\ldots ,\gamma -1\} \\ j_2-j_1\ge 2 \end{array}}. \end{aligned}$$

Proof

We use the notations and set-up from Sect. 2.3. Theorems 10 and 15 imply that there exist divisors $D_\ell \sim 2H-B_\ell R$ on ${\mathbb {P}}({\mathcal {E}})$, with $\ell \in \{2,\ldots ,\gamma -2\}$, and divisors $D_{j_1,j_2}\sim 2H-B_{j_1,j_2}R$ on ${\mathbb {P}}({\mathcal {E}})$, with $j_1,j_2\in \{1,\ldots ,\gamma -1\}$ and $j_2-j_1\ge 2$, such that $C'$ is the scheme-theoretical intersection of these divisors. So we can use Theorem 1 to conclude the proof. Note that indeed

$$\begin{aligned} \sum _{\ell \in \{2,\ldots ,\gamma -2\}}\,B_\ell \ + \sum _{\begin{array}{c} j_1,j_2\in \{1,\ldots ,\gamma -1\} \\ j_2-j_1\ge 2 \end{array}}\,B_{j_1,j_2} =(\gamma -3)(g-\gamma -1), \end{aligned}$$

as announced in the theorem. $\square$

We believe that the above theorem is of independent interest. For instance it is not well-understood which sets of (first) scrollar Betti numbers are possible for canonical curves of a given genus and gonality, and our result can be used to prove certain existence results. It has been conjectured that “most” curves have so-called balanced (first) scrollar Betti numbers, meaning that $\max |b_i-b_j|\le 1$, see [1] and the references therein. Non-degenerate curves are typically highly non-balanced, since one expects the $B_{j_1,j_2}$’s to be about twice the $B_\ell$’s.

Example 17

Consider the following lattice polygons $\Delta _1$ and $\Delta _2$ of lattice width 7 (and lattice-width direction $v=(1,0)$), which only differ from each other at the right hand side.

The polygon $\Delta _1$ does not satisfy all the combinatorial constraints of Theorem 16, since condition ${\mathcal {P}}_2(v)$ does not hold: ${\mathcal {S}}_{2,6}^{(-)}=\{1\}$ and ${\mathcal {S}}_{2,6}^{(+)}=\{2\}$. Although we have not pursued this, we believe that the conditions ${\mathcal {P}}_1(v)$ and ${\mathcal {P}}_2(v)$ are always fulfilled if $\gamma <7$.

On the other hand, the polygon $\Delta _2$ meets all the conditions from the statement, and so we can apply Theorem 16. The first scrollar Betti numbers of a $\Delta _2$-degenerate curve are as follows:

$$\begin{aligned} \begin{array}{lllll} B_{1,3}=12 &{}\quad B_{1,4}=10 &{}\quad B_{1,5}=10 &{}\quad B_{1,6}=9 &{}\quad B_{2,4}=10 \\ B_{2,5}=10 &{}\quad B_{2,6}=9 &{}\quad B_{3,5}=8 &{}\quad B_{3,6}=8 &{}\quad B_{4,6}=7 \\ B_2=4 &{}\quad B_3=4 &{}\quad B_4=4 &{}\quad B_5=3 &{} \end{array} \end{aligned}$$

The sum of these numbers is 108, which agrees with $(\gamma -3)(g-\gamma -1)$ for $g=35$ and $\gamma =7$.

3 Intrinsicness on ${\mathbb {P}}^1\times {\mathbb {P}}^1$

Theorem 18

Let $f \in k[x^{\pm 1},y^{\pm 1}]$ be non-degenerate with respect to its (two-dimensional) Newton polygon $\Delta = \Delta (f)$, and assume that $\Delta \not \cong 2\Upsilon$. Then U(f) is birationally equivalent to a smooth projective genus g curve in ${\mathbb {P}}^1 \times {\mathbb {P}}^1$ if and only if $\Delta ^{(1)} = \emptyset$ or $\Delta ^{(1)} \cong \square _{a,b}$ for some integers $a\ge b \ge 0$, necessarily satisfying $g=(a+1)(b+1)$.

Proof

We may assume that U(f) is neither rational, nor elliptic or hyperelliptic (and hence that $\Delta ^{(1)}$ is two-dimensional) because such curves admit smooth complete models in ${\mathbb {P}}^1 \times {\mathbb {P}}^1$. So for the ‘if’ part we can assume that $b \ge 1$. But then $\text {Tor}(\Delta ^{(1)}) \cong {\mathbb {P}}^1 \times {\mathbb {P}}^1$, and the statement follows using the canonical embedding (3).

The real deal is the ‘only if’ part. At least, if a curve C / k is birationally equivalent to a (non-rational, non-elliptic, non-hyperelliptic) smooth projective curve in ${\mathbb {P}}^1 \times {\mathbb {P}}^1$, then it is $\Delta '$-non-degenerate with $\Delta ' = [-1, a+1] \times [-1, b+1]$ for $a \ge b \ge 1$: this follows from the material in [3, Section 4] (one can use an automorphism of ${\mathbb {P}}^1 \times {\mathbb {P}}^1$ to ensure appropriate behavior with respect to the toric boundary). Note that $\Delta '^{(1)} = \square _{a,b}$. The geometric genus of C equals $g=(a+1)(b+1)$ by [8] and its gonality equals $\gamma = b + 2$ by [3, Cor. 6.2]. We observe that

• g is composite.

• C has Clifford dimension equal to 1 by [3, Theorem 8.1].

• the scrollar invariants of C (with respect to any gonality pencil) are all equal to $g/(\gamma - 1) - 1$. Indeed, by [3, Theorem 6.1], every gonality pencil is computed by projecting along some lattice-width direction v. If $a > b$, then the only pair of lattice-width directions is $\pm (1, 0)$ and from [3, Theorem 9.1], we find that the corresponding scrollar invariants are $a, a, \dots , a$. If $a = b$, we also have the pair $\pm (0,1)$, giving rise to the same scrollar invariants.

• if $\gamma \ge 4$, then the first scrollar Betti numbers (with respect to any gonality pencil) take exactly two distinct values: $2g/(\gamma - 1) - 2$ and $g/(\gamma - 1) - 3$. Indeed, $\Delta '$ satisfies condition ${\mathcal {P}}_1(v)$ (see Example 8), but also condition ${\mathcal {P}}_2(v)$: take $(j_1,j_2)=(j,\ell )$ if $j\in \{-1,\ldots ,b+1\}$, $(j_1,j_2)=(j+1,\ell -1)$ if $j=-1$ and $(j_1,j_2)=(j-1,\ell +1)$ if $j=b+1$. By Theorem 16 we find that these numbers are $2a, 2a, \dots , 2a, a-2,a-2, \dots , a-2$.

A first consequence is that U(f) admits a combinatorial gonality pencil. Indeed, $\Delta$ cannot be of the form $2\Upsilon$ (excluded in the statement of the theorem), nor of the form $d\Sigma$ for some $d \ge 2$: the cases $d =2$ and $d=3$ correspond to rational and elliptic curves (excluded at the beginning of this proof), the case $d=4$ corresponds to curves of genus 3 (not composite), and the cases where $d \ge 5$ correspond to curves of Clifford dimension 2.

Without loss of generality we may then assume that $v=(1,0)$ and $\Delta \subset \{ \, (i,j)\in {\mathbb {R}}^2\,|\, 0\le j\le \gamma \, \}$, so that our gonality pencil corresponds to the vertical projection. By [3, Theorem 9.1], the numbers $E_\ell = -1+\sharp \{(i,j)\in \Delta ^{(1)}\cap {\mathbb {Z}}^2\,|\,j=\ell \}$ (for $\ell = 1, \dots , \gamma - 1$) are the corresponding scrollar invariants. Hence the $E_\ell$’s must all be equal to $E:=g/(\gamma - 1) - 1\ge 1$.

This already puts severe restrictions on the possible shapes of $\Delta ^{(1)}$. By horizontally shifting and skewing we may assume that the lattice points at height $j=1$ are $(0,1), \dots , (E, 1)$ and that the lattice points at height $j=2$ are $(0,2), \dots , (E,2)$. If $\gamma = 3$, it follows that $\Delta ^{(1)}$ has the desired rectangular shape, so we may suppose that $\gamma \ge 4$. Then by horizontally flipping if needed, we can assume that the lattice points at height $j=3$ are $(i,3), \dots , (E + i, 3)$ for some $i \ge 0$. Now $i \ge 2$ is impossible, for this would introduce a new lattice point at height $j=2$; thus $i = 0$ or $i=1$. Continuing this type of reasoning, we obtain that the lattice points of $\Delta ^{(1)}$ can be seen as a pile of n blocks of respectively $m_1, \dots , m_n$ sheets, where each block is shifted to the right over a distance 1 when compared to its predecessor (Fig. 7).

Fig. 7

Lattice points of $\Delta ^{(1)}$ in sheets

We need to show that $n = 1$, because then $\Delta ^{(1)}$ has the desired rectangular shape. We will first prove that $\Delta$ statisfies condition ${\mathcal {P}}_1(v)$. Since $i^{(+)}(j)-i^{(-)}(j)=E$ for each value of j, the inequality

$$\begin{aligned} \epsilon ^{(-)}_{j_1,j_2,r}+\epsilon ^{(+)}_{j_1,j_2,r}\le 1 \end{aligned}$$

holds (so never $\epsilon ^{(-)}_{j_1,j_2,r}=\epsilon ^{(+)}_{j_1,j_2,r}=1$) for all $j_1,j_2\in \{2,\ldots ,\gamma -2\}$ with $j_2-j_1\ge 2$ and $r\in \left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\}$. This implies that

$$\begin{aligned} {\mathcal {S}}^{(-)}_{j_1,j_2}\cup {\mathcal {S}}^{(+)}_{j_1,j_2}=\left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\} . \end{aligned}$$

Now assume that ${\mathcal {S}}^{(-)}_{j_1,j_2}$ and ${\mathcal {S}}^{(+)}_{j_1,j_2}$ are non-empty and disjoint. In this case, we can take $r,s\in \left\{ 1,\ldots ,\left\lfloor \frac{j_2-j_1}{2}\right\rfloor \right\}$ such that

$$\begin{aligned} \epsilon ^{(-)}_{j_1,j_2,r}=\epsilon ^{(+)}_{j_1,j_2,s}=0 \quad \text { and } \quad \epsilon ^{(+)}_{j_1,j_2,r}=\epsilon ^{(-)}_{j_1,j_2,s}=1. \end{aligned}$$

If $r<s$, we get that

$$\begin{aligned} i^{(-)}(j_1 + s) + i^{(-)}(j_2 - s) > i^{(-)}(j_1) + i^{(-)}(j_2) \end{aligned}$$

and

$$\begin{aligned} i^{(+)}(j_1) + i^{(+)}(j_2) > i^{(+)}(j_1 + r) + i^{(+)}(j_2 - r). \end{aligned}$$

Subtracting E from both sides of the latter equation yields

$$\begin{aligned} i^{(-)}(j_1) + i^{(-)}(j_2) > i^{(-)}(j_1 + r) + i^{(-)}(j_2 - r), \end{aligned}$$

so

$$\begin{aligned}&i^{(-)}((j_1+r)+(s-r))+i^{(-)}((j_2-r)-(s-r)) \\&\quad \ge i^{(-)}(j_1+r)+i^{(-)}(j_2-r)+2, \end{aligned}$$

which is in contradiction with Lemma 2. A similar contradiction can be obtained if $r>s$.

Now let’s prove that $\Delta$ also satisfies property ${\mathcal {P}}_2(v)$, where we assume that $n\ge 2$. By Remark 13 and a symmetry consideration (rotation over $180^\circ$), it suffices to check the condition for lattice points (i, j) that lie on the left side of the boundary of $\Delta$ (and even of $\Delta ^{max}$). Take $\ell \in \{2,\ldots ,\gamma -2\}$, $w=(i^{(--)}(\ell ),\ell )$ and $u_{i,j}=(i_1,j_1),v_{i,j}=(i_2,j_2)\in \Delta ^{(1)}$ such that $(i,j)-w=(u_{i,j}-w)+(v_{i,j}-w)$, hence $j_1+j_2=j+\ell$. It is sufficient to prove that

$$\begin{aligned} i+i^{(++)}(\ell )\le i^{(+)}(j_1)+i^{(+)}(j_2). \end{aligned}$$

(10)

First assume that $|j-\ell |>|j_2-j_1|$. If $j\in \{1,\ldots ,\gamma -1\}$, then

$$\begin{aligned} (i+E)+i^{(++)}(\ell )\le & {} (i^{(-)}(j)+E)+i^{(+)}(\ell ) \\= & {} i^{(+)}(j)+i^{(+)}(\ell ) \\\le & {} i^{(+)}(j_1)+i^{(+)}(j_2)+1, \end{aligned}$$

where we use Lemma 2 for the last inequality. Since $E\ge 1$, the desired inequality (10) follows. We still need to check (10) for points $(i,j)\in \partial \Delta$ with $j=0$ and $j=\gamma$, in particular $i=-1$ resp. $i=n-1$ because we can assume that (i, j) lies on the left side of the boundary of $\Delta ^{max}$.

• If $(i,j)=(-1,0)$, the line segment L between $(i+E,j)=(E-1,0)$ and $w'=(i^{(++)}(\ell ),\ell )\in \Delta ^{(2)}$ intersects the horizontal lines of heights $j_1$ and $j_2$ in points that belong to $\Delta ^{(1)}$. Using a similar argument as in the proof of Lemma 2, we obtain that $(i+E)+i^{(++)}(\ell )\le i^{(+)}(j_1)+i^{(+)}(j_2)+1$, which gives us (10) using $E\ge 1$.

• Analogously, we can handle the case $(i,j)=(n-1,\gamma )$: the line segment L between $(i+E,j)=(n+E-1,\gamma )$ and $w'$ will intersect the horizontal lines of heights $j_1$ and $j_2$ in points that are contained in $\Delta ^{(1)}$ and (10) follows.

If $|j-\ell |=|j_2-j_1|$, we may assume that $j_1=j\in \{1,\ldots ,\gamma -1\}$ and $j_2=\ell \in \{2,\ldots ,\gamma -2\}$. But then the inequalities $i\le i^{(-)}(j_1)\le i^{(+)}(j_1)$ and $i^{(++)}(\ell )\le i^{(+)}(j_2)$ yield (10).

We still have to consider the case where $|j-\ell |<|j_2-j_1|$, which implies that $j\in \{1,\ldots ,\gamma -1\}$. By Lemma 2, we have that

$$\begin{aligned} i^{(-)}(j)+i^{(-)}(\ell )\le i^{(-)}(j_1)+i^{(-)}(j_2)+1, \end{aligned}$$

hence

$$\begin{aligned} i+i^{(++)}(\ell )\le & {} i^{(-)}(j)+i^{(+)}(\ell ) \\= & {} i^{(-)}(j)+i^{(-)}(\ell )+E \\\le & {} i^{(-)}(j_1)+i^{(-)}(j_2)+E+1 \\\le & {} i^{(+)}(j_1)+i^{(+)}(j_2). \end{aligned}$$

Since both conditions ${\mathcal {P}}_1(v)$ and ${\mathcal {P}}_2(v)$ hold for $\Delta$, we can apply Theorem 16. If $n \ge 2$, then there is at least one first scrollar Betti number having value $E - 1 = g/(\gamma - 1) - 2$, for instance $B_2$. This is distinct from both $2g/(\gamma - 1) - 2$ and $g/(\gamma - 1) - 3$: contradiction. Therefore $n=1$, i.e. $\Delta ^{(1)}$ has the requested rectangular shape. $\square$

4 Open questions

Here are two interesting open questions related to this paper:

• 1. In Sect. 2, we gave a combinatorial interpretation for the first scrollar Betti numbers of $\Delta$-non-degenerate curves C in terms of the combinatorics of $\Delta$, in case $\Delta$ satisfies the condition ${\mathcal {P}}_1(v)$ (see Definition 4) and ${\mathcal {P}}_2(v)$ (see Definition 12). Can this be generalized to all polygons $\Delta$? There seems to be no geometric reason why this wouldn’t be the case, but we did not succeed to get rid of the conditions.

• 2. In Theorem 18 of Sect. 3, we showed that non-degenerate curves on ${\mathbb {P}}^1\times {\mathbb {P}}^1$ have an intrinsic Newton polygon (at least, if $g\ne 4$). Can this be generalized to $\Delta$-non-degenerate curves on Hirzebruch surfaces ${\mathcal {H}}_n$? In this case, we expect $\Delta ^{(1)} = \emptyset$ or $\Delta ^{(1)} \cong \text {conv}\{(0,0),(a+nb,0),(a,b),(0,b)\}$ for some integers $a,b,n \ge 0$ .