Computing minimal generating systems for some special toric ideals

TL;DR

This paper discusses a method for efficiently computing minimal generating systems for certain toric ideals associated with lattice polytopes, simplifying the process to Gaussian elimination when boundary lattice points are at least four.

Contribution

It provides a straightforward approach to determine minimal generators of toric ideals for specific projective toric surfaces, reducing the problem to linear algebra techniques.

Findings

Minimal generating systems can be computed via Gaussian elimination.

Applicable when the boundary of the lattice polytope has at least four lattice points.

Simplifies the algebraic process for certain toric ideals.

Abstract

Let $X_{P}$ be the projective toric surface associated to a lattice polytope $P$. If the number of lattice points lying on the boundary of $P$ is at least $4$, it is known that $X_{P}$ is embeddable into a suitable projective space as zero set of finitely many quadrics. In this case, the determination of a minimal generating system of the toric ideal defining $X_{P}$ is reduced to a simple Gaussian elimination.

Equations

$$P=\left\{\mathbf{x}\in {\mathbb{R}}^2|⟨\mathbf{x},{\mathbf{u}}_F⟩\ge -a_F\text{ for all }F\in \mathcal{F}(P)\right\},$$

$$D_P:=\sum _{F\in \mathcal{F}(P)}{a}_F\overline{\text{orb}({\mathbb{R}}_{\ge 0}{\mathbf{u}}_F)}$$

$$\mathbb{T}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbb{T}\ni t⟼({\Phi }_{|D_P|}\circ \iota )(t):=[...:z_{(i,j)}:...{]}_{(i,j)\in P\cap {\mathbb{Z}}^2}\in {\mathbb{P}}_{\mathbb{C}}^{{\delta }_P},z_{(i,j)}:={\chi }^{(i,j)}(t),$$

$$S_{P}:=\mathbb{C}[C(P)\cap\mathbb{Z}^{3}]={\displaystyle\bigoplus\limits_{\kappa=0}^{\infty}}\left({\displaystyle\bigoplus\limits_{(i,j)\in(\kappa P)\cap\mathbb{Z}^{2}}}\mathbb{C\cdot}\chi^{(i,j)}s^{\kappa}\right)\$$

$${\mathcal{A}}_P:=\left\{(i,j,1)|(i,j)\in P\cap {\mathbb{Z}}^2\right\}\subset {\mathbb{Z}}^2\times \{1\}\subset {\mathbb{Z}}^3,$$

$$\mathbb{C}[...:z_{(i,j)}:....{]}_{(i,j)\in P\cap {\mathbb{Z}}^2}\stackrel{{\pi }_P}{⟶}\mathbb{C}[...,{\chi }^{(i,j,1)},....{]}_{(i,j,1)\in {\mathcal{A}}_P},z_{(i,j)}⟼{\chi }^{(i,j,1)}.$$

$$I_{\mathcal{A}_{P}}=\left\langle\left\{z_{(i_{1},j_{1})}z_{(i_{2},j_{2})}-z_{(i_{1}^{\prime},j_{1}^{\prime})}z_{(i_{2}^{\prime},j_{2}^{\prime})}\left|\begin{array}[c]{c}(i_{1},j_{1}),(i_{2},j_{2}),(i_{1}^{\prime},j_{1}^{\prime}),(i_{2}^{\prime},j_{2}^{\prime})\in P\cap\mathbb{Z}^{2},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{\emph{with} }(i_{1},j_{1})+(i_{2},j_{2})=(i_{1}^{\prime},j_{1}^{\prime})+(i_{2}^{\prime},j_{2}^{\prime})\end{array}\right.\right\}\right\rangle.$$

$${\beta }_P=\left(\genfrac{}{}{0ex}{}{{\delta }_P+2}{2}\right)-♯(2P\cap {\mathbb{Z}}^2).$$

$$f:{\text{HP}}_2({\mathbb{P}}_{\mathbb{C}}^{{\delta }_P})⟶\mathbb{C}\left[x^{\pm 1},y^{\pm 1}\right],\text{ mapping }{z}_{(i_1,j_1)}{z}_{(i_2,j_2)}\text{ onto }{x}^{i_1+i_2}{y}^{j_1+j_2},$$

$$P_{a,b}:=\text{conv}(\{(0,0),(a,0),(b,1),(0,1)\})$$

$${\beta }_{P_{a,b}}$$

$$=\frac{(a+b+2)(a+b+3)}{2}-3(a+b+1)=\frac{1}{2}\left(a+b-1\right)\left(a+b\right)$$

$$\text{rank}\left(\begin{array}[c]{cccccccc}z_{(0,0)}&z_{(1,0)}&\cdots&z_{(a-1,0)}&z_{(0,1)}&z_{(1,1)}&\cdots&z_{(b-1,1)}\\ z_{(1,0)}&z_{(2,0)}&\cdots&z_{(a,0)}&z_{(1,1)}&z_{(2,1)}&\cdots&z_{(b,1)}\end{array}\right)\leq 1.$$

$$\mathbb{C}[{\mathsf{X}}_0,{\mathsf{X}}_1,{\mathsf{X}}_2{]}_d:=\left\{F\in \mathbb{C}[{\mathsf{X}}_0,{\mathsf{X}}_1,{\mathsf{X}}_2]|F\text{ homogeneous of degree }d\right\}\cup \{0\}$$

$${\mathcal{E}}_{2,d}:=\left\{\mathbf{\alpha }=({\alpha }_0,{\alpha }_1,{\alpha }_2)\in {\mathbb{Z}}^3|{\alpha }_0,{\alpha }_1,{\alpha }_2\in [0,d]\text{ and }{\alpha }_0+{\alpha }_1+{\alpha }_2=d\right\}.$$

$${\text{Tr}}_d:=\text{conv}(\{(0,0),(d,0),(0,d)\})=\left\{(x_1,x_2)\in {\mathbb{R}}^2|x_1\ge 0,x_2\ge 0\text{ and }{x}_1+x_2\le d\right\}$$

$${\text{Tr}}_d\cap {\mathbb{Z}}^2\ni (m_1,m_2)⟼(m_1+1,m_1+m_2+2)\in \left\{({\xi }_1,{\xi }_2)\in {\mathbb{Z}}_{\ge 0}^2|1\le {\xi }_1<{\xi }_2\le d+2\right\}$$

$$\begin{array}[c]{r}\sharp(\mathcal{E}_{2,d})=\sharp\left\{(\alpha_{0},\alpha_{1},\alpha_{2})\in\mathbb{Z}^{3}\left|\alpha_{0},\alpha_{1},\alpha_{2}\in[0,d]\text{ and }\alpha_{0}+\alpha_{1}+\alpha_{2}\leq d\right.\right\}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ -\sharp\left\{(\alpha_{0},\alpha_{1},\alpha_{2})\in\mathbb{Z}^{3}\left|\alpha_{0},\alpha_{1},\alpha_{2}\in[0,d-1]\text{ and }\alpha_{0}+\alpha_{1}+\alpha_{2}\leq d-1\right.\right\}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ =\tbinom{d+3}{3}-\tbinom{d+2}{3}=\tbinom{d+2}{2}=\sharp(\text{Tr}_{d}\cap\mathbb{Z}^{2}).\end{array}$$

$$X_{{\text{Tr}}_d}\cong \mathbb{V}(I_{{\mathcal{A}}_{{\text{Tr}}_d}})\subset {\mathbb{P}}_{\mathbb{C}}^{{\delta }_{{\text{Tr}}_d}},\text{ with }{\delta }_{{\text{Tr}}_d}=\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)-1,\text{ and}$$

$$I_{\mathcal{A}_{\text{Tr}_{d}}}=\left\langle\left\{z_{(i_{1},j_{1})}z_{(i_{2},j_{2})}-z_{(i_{1}^{\prime},j_{1}^{\prime})}z_{(i_{2}^{\prime},j_{2}^{\prime})}\left|\begin{array}[c]{c}(i_{1},j_{1}),(i_{2},j_{2}),(i_{1}^{\prime},j_{1}^{\prime}),(i_{2}^{\prime},j_{2}^{\prime})\in\text{Tr}_{d}\cap\mathbb{Z}^{2},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{with }(i_{1},j_{1})+(i_{2},j_{2})=(i_{1}^{\prime},j_{1}^{\prime})+(i_{2}^{\prime},j_{2}^{\prime})\end{array}\right.\right\}\right\rangle,$$

$${\nu }_{2,d}:{\mathbb{P}}_{\mathbb{C}}^2\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^{{\delta }_{{\text{Tr}}_d}},[{\mathsf{X}}_0:{\mathsf{X}}_1:{\mathsf{X}}_2]⟼[...:{\mathsf{X}}^{\mathbf{\alpha }}:....{]}_{\mathbf{\alpha }\in {\mathcal{E}}_{2,d}}.$$

$${\beta }_{{\text{Tr}}_d}$$

$$=\frac{1}{2}\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)((\genfrac{}{}{0ex}{}{d+2}{2})+1)-(2d^2+3d+1)$$

$$=\frac{(d+1)(d+2)}{4}(\frac{(d+1)(d+2)}{2}+1)-(2d^2+3d+1)=\frac{1}{8}d\left(d+6\right)\left(d^2-1\right).$$

$$B=\{z_{(i,j)}{z}_{(i^{\prime },j^{\prime })}:(i,j),(i^{\prime },j^{\prime })\in P\cap {\mathbb{Z}}^2\}$$

$$(i_1,j_1)+(i_2,j_2)=(i_1^{\prime },j_1^{\prime })+(i_2^{\prime },j_2^{\prime })$$

$$\begin{array}[c]{ll}z_{(0,0)}z_{(2,0)}-z_{(1,0)}^{2},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,0)}z_{(0,2)}-z_{(0,1)}^{2},\\ z_{(0,0)}z_{(1,1)}-z_{(0,1)}z_{(1,0)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,1)}z_{(1,1)}-z_{(0,2)}z_{(1,0)},\\ z_{(0,2)}z_{(2,0)}-z_{(1,1)}^{2},&z_{(0,1)}z_{(2,0)}-z_{(1,0)}z_{(1,1)}.\end{array}$$

$$\begin{array}[c]{lll}z_{(1,1)}z_{(3,0)}-z_{(2,0)}z_{(2,1)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,3)}z_{(2,1)}-z_{(1,2)}^{2},&z_{(0,0)}z_{(1,2)}-z_{(0,2)}z_{(1,0)},\\ z_{(0,3)}z_{(3,0)}-z_{(1,2)}z_{(2,1)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,1)}z_{(3,0)}-z_{(1,0)}z_{(2,1)},&z_{(0,1)}z_{(2,1)}-z_{(1,1)}^{2},\\ z_{(0,0)}z_{(2,0)}-z_{(1,0)}^{2},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,2)}z_{(3,0)}-z_{(1,1)}z_{(2,1)},&z_{(0,2)}z_{(2,1)}-z_{(0,3)}z_{(2,0)},\\ z_{(0,2)}z_{(2,1)}-z_{(1,1)}z_{(1,2)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,2)}z_{(3,0)}-z_{(1,2)}z_{(2,0)},&z_{(0,1)}z_{(1,2)}-z_{(0,2)}z_{(1,1)},\\ z_{(0,0)}z_{(2,1)}-z_{(1,0)}z_{(1,1)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,0)}z_{(2,1)}-z_{(0,1)}z_{(2,0)},&z_{(0,1)}z_{(2,1)}-z_{(0,2)}z_{(2,0)},\\ z_{(0,0)}z_{(3,0)}-z_{(1,0)}z_{(2,0)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,0)}z_{(0,2)}-z_{(0,1)}^{2},&z_{(0,0)}z_{(1,1)}-z_{(0,1)}z_{(1,0)},\\ z_{(0,1)}z_{(0,3)}-z_{(0,2)}^{2},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(1,0)}z_{(3,0)}-z_{(2,0)}^{2},&z_{(0,1)}z_{(1,2)}-z_{(0,3)}z_{(1,0)},\\ z_{(0,2)}z_{(1,2)}-z_{(0,3)}z_{(1,1)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(1,2)}z_{(3,0)}-z_{(2,1)}^{2},&z_{(0,1)}z_{(2,1)}-z_{(1,0)}z_{(1,2)},\\ z_{(0,0)}z_{(0,3)}-z_{(0,1)}z_{(0,2)},\vskip 6.0pt plus 2.0pt minus 2.0pt&z_{(0,1)}z_{(3,0)}-z_{(1,1)}z_{(2,0)},&z_{(0,0)}z_{(1,2)}-z_{(0,1)}z_{(1,1)}\end{array}$$

$$\ell :=\min \{\kappa \in {\mathbb{Z}}_{>0}|\mathcal{V}(\kappa \mathring{Q})\subset {\mathbb{Z}}^2\}.$$

$$Q:=\text{ conv}\left(\left\{(0,1),(1,1),(1,0),(0,-1),(-1,-1),(-1,0)\right\}\right)$$

$$\mathring{Q}:=\text{conv}\left(\left\{(1,0),(1,-1),(0,-1),(-1,0),(-1,1),(0,1)\right\}\right)$$

Full text

1. Introduction

Let $P\subset\mathbb{R}^{2}$ be a lattice polygon, i.e., a (convex, 2-dimensional) polygon, all of whose vertices belong to $\mathbb{Z}^{2}.$ $P$ is known to be normal and very ample, and to have a canonical presentation

[TABLE]

where $\mathcal{F}(P)$ is the set of the facets (edges) of $P,$ $a_{F}\in\mathbb{Z}$ and $\mathbf{u}_{F}\in\mathbb{Z}^{2}$ the inward-pointing facet normal, i.e., the minimal generator of the ray $\mathbb{R}_{\geq 0}\mathbf{u}_{F}.$ The corresponding compact complex toric surface $X_{P}$ is therefore normal and projective, and

[TABLE]

is a very ample Cartier divisor on $X_{P},$ where $\overline{\text{orb}(\mathbb{R}_{\geq 0}\mathbf{u}_{F})}$ is the Zariski closure of the orbit of the ray $\mathbb{R}_{\geq 0}\mathbf{u}_{F}$ w.r.t. the natural Hom${}_{\mathbb{Z}}(\mathbb{Z}^{2},\mathbb{C}^{\ast})$-action. (See [4, Corollaries 2.2.13 and 2.2.19 (b), pp. 70-71, (4.2.6), p. 182, and Proposition 6.1.10 (c), p. 269 ].) Setting $\delta_{P}:=\sharp(P\cap\mathbb{Z}^{2})-1,$ the complete linear system $\left|D_{P}\right|$ induces the closed embedding $\Phi_{\left|D_{P}\right|}$,

[TABLE]

with

[TABLE]

where $\chi^{(i,j)}:\mathbb{T}\rightarrow\mathbb{C}^{\ast}$ is the character associated to the lattice point $(i,j)$ (with $\mathbb{T}$ denoting the algebraic torus Hom${}_{\mathbb{Z}}(\mathbb{Z}^{2},\mathbb{C}^{\ast})$), for all $(i,j)\in P\cap\mathbb{Z}^{2}.$ The image $\Phi_{\left|D_{P}\right|}(X_{P})$ of $X_{P}$ under $\Phi_{\left|D_{P}\right|}$ is the Zariski closure of Im$(\Phi_{\left|D_{P}\right|}\circ\iota)$ in $\mathbb{P}_{\mathbb{C}}^{\delta_{P}}$ and can be viewed as the projective variety Proj$(S_{P}),$ where

[TABLE]

$($with $C(P):=\{(\lambda y_{1},\lambda y_{2},\lambda)\left|\lambda\in\mathbb{R}_{\geq 0}\text{ and }(y_{1},y_{2})\in P\right.\})$ is the semigroup algebra which is naturally graded by setting deg$(\chi^{(i,j)}s^{\kappa}):=\kappa.$ (For a detailed exposition see [4, Theorem 2.3.1, p. 75; Proposition 5.4.7, pp. 237-238; Theorem 5.4.8, pp. 239-240, and Theorem 7.1.13, pp. 325-326].) Equivalently, it can be viewed as the zero set $\mathbb{V}(I_{\mathcal{A}_{P}})\subset\mathbb{P}_{\mathbb{C}}^{\delta_{P}}$ of the homogeneous ideal $I_{\mathcal{A}_{P}}:=$ Ker$(\pi_{P}),$ where

[TABLE]

and $\pi_{P}$ is the $\mathbb{C}$-algebra homomorphism

[TABLE]

Theorem 1.1** (Koelman [7]).**

If $\sharp(\partial P\cap\mathbb{Z}^{2})\geq 4,$ then $I_{\mathcal{A}_{P}}$ is generated by all possible quadratic binomials, i.e.,

[TABLE]

Corollary 1.2** (Castryck & Cools [2, §2]).**

If $\sharp(\partial P\cap\mathbb{Z}^{2})\geq 4,$ and if we denote by $\beta_{P}$ the cardinality of any minimal system of quadrics generating the ideal $I_{\mathcal{A}_{P}},$ then

[TABLE]

Proof.

If HP${}_{2}(\mathbb{P}_{\mathbb{C}}^{\delta_{P}}):=\left\{\text{homogeneous polynomials (in }\delta_{P}+1\text{ variables) of degree }2\right\},$ then the $\mathbb{C}$-vector space homomorphism

[TABLE]

has as kernel Ker$(f)$ the $\mathbb{C}$-vector space of homogeneous polynomials of degree $2$ which belong to $I_{\mathcal{A}_{P}}$ and as image Im$(f)$ the linear span of $\left.\{x^{i}y^{j}\right|(i,j)\in 2P\cap\mathbb{Z}^{2}\}$ (because every lattice point in $2P$ is the sum of two lattice points of $P,$ cf. [4, Theorem 2.2.12, pp. 68-69]). Taking into account Koelman’s Theorem 1.1, [8, Lemma 4.1, p. 31], and the fact that $\mathbb{V}(I_{\mathcal{A}_{P}})$ is not contained in any hyperplane of $\mathbb{P}_{\mathbb{C}}^{\delta_{P}},$ the equality $\dim_{\mathbb{C}}(\text{Ker}(f))=\dim_{\mathbb{C}}(\text{HP}_{2}(\mathbb{P}_{\mathbb{C}}^{\delta_{P}}))-\dim_{\mathbb{C}}(\operatorname{Im}(f))$ gives (1.1). ∎

Examples 1.3**.**

(i) If $a,b$ are two positive integers, then the projective toric surface $X_{P_{a,b}}\cong\mathbb{V}(I_{\mathcal{A}_{P_{a,b}}})\subset\mathbb{P}_{\mathbb{C}}^{\delta_{P}}$ which is associated to the lattice quadrilateral

[TABLE]

(where “conv” stands for convex hull, and $\delta_{P_{a,b}}=a+b+1$), is isomorphic to the intersection of

[TABLE]

quadrics, i.e., to the rational normal scroll of type $(a,b)$ w.r.t. the homogeneous coordinates $[...:z_{(i,j)}:....]_{(i,j)\in P_{a,b}\cap\mathbb{Z}^{2}}$ satisfying the “2-minors condition”

[TABLE]

In particular, for $a=b=1,$ $X_{P_{1,1}}\cong\mathbb{V}(z_{(0,0)}z_{(1,1)}-z_{(1,0)}z_{(0,1)})\subset\mathbb{P}_{\mathbb{C}}^{3}$ can be viewed as the classical (smooth) quadric hypersurface in $\mathbb{P}_{\mathbb{C}}^{3}$ (which is isomorphic to $\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$ and birationally equivalent to $\mathbb{P}_{\mathbb{C}}^{2},$ cf. Figure 1).

(ii) Let $d$ be a positive integer. The $\mathbb{C}$-vector space

[TABLE]

has the set $\{\mathsf{X}_{0}^{\alpha_{0}}\mathsf{X}_{1}^{\alpha_{1}}\mathsf{X}_{2}^{\alpha_{2}}\left|(\alpha_{0},\alpha_{1},\alpha_{2})\in\mathcal{E}_{2,d}\right.\}$ as one of its bases, where

[TABLE]

For each $\boldsymbol{\alpha}=(\alpha_{0},\alpha_{1},\alpha_{2})\in\mathcal{E}_{2,d}$ we write $\mathsf{X}^{\boldsymbol{\alpha}}:=\mathsf{X}_{0}^{\alpha_{0}}\mathsf{X}_{1}^{\alpha_{1}}\mathsf{X}_{2}^{\alpha_{2}}.$ Setting

[TABLE]

we see that $\sharp(\partial$Tr${}_{d}\cap\mathbb{Z}^{2})=3d$ and $\sharp($Tr${}_{d}\cap\mathbb{Z}^{2})=\tbinom{d+2}{2},$ because

[TABLE]

is a bijective map with $\sharp\left\{(\xi_{1},\xi_{2})\in\mathbb{Z}_{\geq 0}^{2}\left|\text{ }1\leq\xi_{1}<\xi_{2}\leq d+2\right.\right\}=\tbinom{d+2}{2}.$ On the other hand,

[TABLE]

If $d\geq 2,$ then using the homogeneous coordinates $[...:z_{(i,j)}:....]_{(i,j)\in\text{Tr}_{d}\cap\mathbb{Z}^{2}}$ we conclude (by Koelman’s Theorem 1.1) that

[TABLE]

[TABLE]

i.e., that $X_{\text{Tr}_{d}}$ is isomorphic to the image of the so-called $d$-uple Veronese embedding

[TABLE]

where the monomials $\left\{\mathsf{X}^{\boldsymbol{\alpha}}\left|\boldsymbol{\alpha}\in\mathcal{E}_{2,d}\right.\right\}$ are arranged in a prescribed manner (e.g., lexicographically). In fact, in this case,

[TABLE]

Note 1.4**.**

For a Magma code for the computation of a minimal generating system of the ideal defining the projective toric surface associated to an arbitrary lattice polygon (and of much more, like Betti numbers etc.), see [3]. In the above mentioned particular case (in which we deal only with quadrics) it is enough (as we shall see in §2) to collect all vectorial relations $(i_{1},j_{1})+(i_{2},j_{2})=(i_{1}^{\prime},j_{1}^{\prime})+(i_{2}^{\prime},j_{2}^{\prime}),$ and to determine a $\mathbb{C}$-linearly independent subset of the set of the corresponding quadratic binomials $z_{(i_{1},j_{1})}z_{(i_{2},j_{2})}-z_{(i_{1}^{\prime},j_{1}^{\prime})}z_{(i_{2}^{\prime},j_{2}^{\prime})}$ by simply performing Gaussian elimination.

2. The algorithm

An algorithm, implemented in Python3 to compute a minimal generating set for the ideal $I_{\mathcal{A}_{P}}$, given the vertex set $\mathcal{V}(P)$ of the polygon $P$ is provided in the library toricIdeal.py. The algorithm is provided by the routine minGenSet, which receives a list of vertices and calls five subroutines to compute a minimal generating set of $I_{\mathcal{A}_{P}}$.

⬇

import numpy

def minGenSet(p):

    intp=integerPoints(*vertToConst(p))

    (basis,genBin)=genBinom(intp)

    indepCol=findIndepCol(genBin)

    binom=findBinom(basis,genBin,indepCol)

    return(intp,binom)

More specifically, the first subroutine, vertToConst, produces a complete system of facet-defining inequalities from the list $\mathcal{V}(P)$ and lower and upper bounds for the coordinates of the points of the polygon. First, the facets are distinguished among the line segments connecting any two vertices, using the fact that the polygon lies entirely in one of the two closed half-planes bounded by their supporting lines and then a constraint is created for each facet.

⬇

def vertToConst(p):

    A=[]

    b=[]

    for i in range(len(p)-1):

            for j in range(i+1,len(p)):

                    big=True

                    small=True

                    coef=numpy.array([(p[i,1]-p[j,1]),

                            (p[j,0]-p[i,0])])

                    cst=numpy.inner(coef,p[j])

                    for point in p:

                            diff=numpy.inner(coef,point)-cst

                            if diff > 0:

                                    small=False

                            elif diff <0:

                                    big=False

            if big :

                    A.append(coef)

                    b.append(cst)

            if small :

                    A.append(-coef)

                    b.append(-cst)

    A=numpy.array(A)

    b=numpy.array(b)

    x=[min(p[:,0]),max(p[:,0])]

    y=[min(p[:,1]),max(p[:,1])]

    return(A,b,x,y)

The second one, integerPoints, uses the constraints and by brute force finds the lattice points of the polygon.

⬇

def integerPoints(A,b,x,y):

    intPoints=[]

    for x0 in range(x[0],x[1]+1):

            for y0 in range(y[0],y[1]+1):

                    point=numpy.array([x0,y0])

                    diff=numpy.dot(A,point)-b

                    inInterior=True

                    for coord in diff:

                            if coord<0:

                                    inInterior=False

                                    break

                    if inInterior:

                            intPoints.append(point)

    intPoints=numpy.array(intPoints)

    return(intPoints)

The third one, genBinom, orders the basis elements

[TABLE]

of the $\mathbb{C}$-vector space $\text{HP}_{2}(\mathbb{P}_{\mathbb{C}}^{\delta_{P}})$ and finds a generating set for the ideal $I_{\mathcal{A}_{P}}$ by collecting all vectorial relations of the form

[TABLE]

where $(i_{1},j_{1}),(i_{2},j_{2}),(i_{1}^{\prime},j_{1}^{\prime}),(i_{2}^{\prime},j_{2}^{\prime})\in P\cap\mathbb{Z}^{2}$. This is returned as a matrix containing the coefficients of the generating binomials w.r.t. the ordered basis $B$.

⬇

def genBinom(intPoints):

    basis=[]

    for i in range(len(intPoints)):

            for j in range(i,len(intPoints)):

                    bp=intPoints[i].tolist()+intPoints[j].tolist()

                    basis.append(bp)

    basis=numpy.array(basis)

    genBin=[]

    for i in range(len(basis)):

            for j in range(i+1,len(basis)):

                    if basis[i,0]+basis[i,2]==basis[j,0]+basis[j,2] and basis[i,1]+basis[i,3]==basis[j,1]+basis[j,3]:

                            row=numpy.zeros(len(basis),

                            dtype=numpy.int)

                            row[i]+=1

                            row[j]-=1

                            genBin.append(row)

    genBin=numpy.transpose(numpy.array(genBin))

    return(basis, genBin)

The fourth one, findIndepCol, performs a Gauss elimination on a matrix and finds a basis of its column space by collecting the non-zero columns of the row echelon form of it.

⬇

from scipy.linalg import lu

def findIndepCol(A):

    U=lu(A,permute_l=True)[1]

    indepCol=[]

    for i in range(len(U)):

            for j in range(len(U[i])):

                    if U[i,j]!=0:

                            indepCol.append(j)

                            break

    return(indepCol)

Finally, the fifth subroutine, findBinom, uses the matrix given by genBinom and the list of $\mathbb{C}$-linearly independent columns found by findIndepCol to produce a set of $\mathbb{C}$-linearly independent generating binomials of the ideal $I_{\mathcal{A}_{P}}$.

⬇

def findBinom(basis,genBin,indepCol):

    binom=[]

    for i in indepCol:

            j1=-1

            j2=-1

            for j in range(len(genBin)):

                    if genBin[j,i]==1:

                            j1=j

                    elif genBin[j,i]==-1:

                            j2=j

                    if j1!=-1 and j2!=-1:

                            break

                    binomial=”z_{{({},{})}}z_{{({},{})}}-z_{{({},{})}}z_{{({},{})}}”.format(basis[j1][0],basis[j1][1],basis[j1][2],basis[j1][3],basis[j2][0],basis[j2][1],basis[j2][2],basis[j2][3])

    binom.append(binomial)

    return(binom)

The complexity of the minGenSet is polynomial of the class $\mathcal{O}(m^{4}+n^{2})$, where $n$ is the number of vertices and $m$ an integer bounding absolutely the coordinates of the vertices.

3. Applications

$\blacktriangleright$ Veronese surfaces. If $P=\text{Tr}_{2}:=\text{conv}(\{(0,0),(2,0),(0,2)\})$ (with $d=2$ as in 1.3 (ii)), then the algorithm produces the following minimal generating set of $I_{\mathcal{A}_{\text{Tr}_{2}}}$:

[TABLE]

Analogously, if $P=\text{Tr}_{3}:=\text{conv}(\{(0,0),(3,0),(0,3)\})$ (with $d=3$), then the $27$ quadrics

[TABLE]

generate minimally $I_{\mathcal{A}_{\text{Tr}_{3}}}$ (cf. (1.2)).

$\blacktriangleright$ Toric log del Pezzo surfaces. These are of the form $X_{P},$ where $P=\ell\mathring{Q}$ is the polar of an LDP-polygon $Q\subset\mathbb{R}^{2}$ dilated by its index $\ell$. (An LDP-polygon $Q\subset\mathbb{R}^{2}$ is a convex polygon which contains the origin in its interior, and its vertices belong to $\mathbb{Z}^{2}$ and are primitive. The index of a polygon of this kind is defined to be

[TABLE]

Kasprzyk, Kreuzer & Nill [6, §6] developed an algorithm by means of which one creates an LDP-polygon, for given $\ell\geq 2,$ by fixing a “special” edge and following a prescribed successive addition of vertices, and produced in this way the long lists of all LDP-polygons for $\ell\leq 17.$ An explicit study for each of these $15346$ LDP-polygons is available on the webpage [1].)

(i) Up to unimodular transformation the only reflexive hexagon (i.e., the only LDP-hexagon of index $1$) is

[TABLE]

having

[TABLE]

as its polar, and $X_{\mathring{Q}}\cong\mathbb{V}(I_{\mathcal{A}_{\mathring{Q}}})\subset\mathbb{P}_{\mathbb{C}}^{6}$ with $I_{\mathcal{A}_{\mathring{Q}}}$ minimally generated by the $9$ quadrics:

[TABLE]

(ii) For the LDP-triangle $Q$ of index $2$ with vertex set

[TABLE]

we obtain

[TABLE]

and $X_{2\mathring{Q}}\cong\mathbb{V}(I_{\mathcal{A}_{2\mathring{Q}}})\subset\mathbb{P}_{\mathbb{C}}^{6}$ with $I_{\mathcal{A}_{2\mathring{Q}}}$ minimally generated by the $7$ quadrics:

[TABLE]

(iii) For the LDP-pentagon $Q$ of index $3$ with vertex set

[TABLE]

(which is unimodularly equivalent to the pentagon “$Q_{2}^{[3]}$” of [5]) we obtain

[TABLE]

and $X_{3\mathring{Q}}\cong\mathbb{V}(I_{\mathcal{A}_{3\mathring{Q}}})\subset\mathbb{P}_{\mathbb{C}}^{38}$ with $I_{\mathcal{A}_{3\mathring{Q}}}$ minimally generated by a set of $646$ quadrics!

(iv) For the LDP-quadrilateral $Q$ of index $4$ with vertex set

[TABLE]

we obtain $\mathcal{V}(4\mathring{Q})=\{(2,-1),(0,-2),(-12,16),(-4,8)\},$ and $X_{4\mathring{Q}}\cong\mathbb{V}(I_{\mathcal{A}_{4\mathring{Q}}})\subset\mathbb{P}_{\mathbb{C}}^{45}$ with $I_{\mathcal{A}_{4\mathring{Q}}}$ minimally generated by a set of $918$ quadrics!

(v) Finally, the LDP-triangle $Q$ of index $5$ with vertex set

[TABLE]

we obtain $\mathcal{V}(5\mathring{Q})=\{(1,-5),(0,-5),(-1,10)\},$ and $X_{5\mathring{Q}}\cong\mathbb{V}(I_{\mathcal{A}_{5\mathring{Q}}})\subset\mathbb{P}_{\mathbb{C}}^{9}$ with $I_{\mathcal{A}_{5\mathring{Q}}}$ minimally generated by the following $21$ quadrics:

[TABLE]