In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P^1 x P^1 in detail and give numerous examples.
In this paper we present an algorithm for computing a matrix representation for a surface in P^3 parametrized over a 2-dimensional toric variety T. This algorithm follows the ideas of [Botbol-Dickenstein-Dohm-09] and it was implemented in Macaulay2. We showed in [BDD09] that such a surface can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection, and in [Botbol-09] we generalized this to the case where the base locus is not necessarily a local complete intersection. The key point consists in exploiting the sparse structu…
This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational maps to smooth toric varieties.
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the correspondence between the irreducible invariant difference subvarieties and the faces of the N[x]-submodules and the orbit-face correspondence. Finally, an algorithm is given to decide whether a binomial di…
We present a matrix-based algorithm for deciding if the parametrization of a curve or a surface is invertible or not, and for computing the inverse of the parametrization if it exists.
We develop representation theory approach to the study of special functions associated with toric varieties. In particular we show that the corresponding special functions are given by matrix elements of certain non-reductive Lie algebras
A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maxi…
Let I be the toric ideal defined by a 2 x n matrix of integers, A = ((1 1 ... 1)(a_1 a_2 ... a_n)) with a_1<a_2<...<a_n. We give a combinatorial proof that I is generated by elements of degree at most the sum of the two largest differences a_i - a_(i-1). The novelty is in the method of proof: the result has already been shown by L'vovsky using cohomological arguments.
This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call \textit{locally toric} a projective curve that in a neighbourhood of every point has a local analytical parametrization of type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in $\bP^3$ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \cite{kaji} by H.Kaji where the same result is proven for any curve in $\bP^3$ such that…
Given a matrix Schubert variety $\overline{X_π}$, it can be written as $\overline{X_π}=Y_π\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_π$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $Φ(\mathbb{P}(Y_π))$ of its projectivization. We construct regular triangulations of $Φ(\mathbb{P}(Y_π))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of…
For a given permutation $π\in S_N$, Fulton proves that the matrix Schubert variety $\overline{X_π} \cong Y_π \times \mathbb{C}^q$ can be defined via certain rank conditions encoded in the Rothe diagram of $π$. In the case where $Y_π:=\text{TV}(σ_π)$ is toric (with respect to a $(\mathbb{C}^*)^{2N-1}$ action), we show that it can be described as an edge ideal of a bipartite graph $G^π$. We characterize the lower dimensional faces of the associated so-called edge cone $σ_π$ explicitly in terms of subgraphs of $G^π$ and present a combinatorial study for the first order deformations of $Y_π$. We p…
We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the $R$-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on the well-known geometric interpretations of matrix model digram techniques, we construct a new model that enumerates triangulations, subdivisions, and numbers of integral points of integral polygons. This approach allows us to look at their combinatorics from a new perspective, motivated by knowledge about matrix models and their integrability. We show how an…
We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of smooth cubic surfaces. The proof is based on a toric degeneration of cubic surfaces, the Givental's mirror theorem for toric manifolds, and the Picard-Lefschetz theory.
Let $\K$ be a finite field and $X$ be a complete simplicial toric variety over $\K$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which can be used to study algebraic geometric codes arising from $Y_Q$. We give a method to compute the lattice $L$ whose ideal $I_L$ is exactly $I(Y_Q)$ under a mild condition. As applications, we give precise descriptions for the lattices corresponding to some special subgroups. We also prove a Nullstellensatz type theorem valid over finite fields, and share \ver…
There exists a covariant non-injective functor from the space of generic Riemann surfaces to the so-called toric AF-algebras; such a functor maps isomorphic Riemann surfaces to the stably isomorphic toric AF-algebras. We use the functor to construct a faithful representation of the mapping class group of surface of genus g>1 into the matrix group GL(6g-6, Z).
The purpose of this article is to show how one might compute the étale cohomology groups $H^p_{\acute{e}t}(X,G_m)$ in degrees $p=0$, $1$ and $2$ of a toric variety $X$ with coefficients in the sheaf of units. The method is to reduce the computation down to the problem of diagonalizing a matrix with integral coefficients. The procedure outlined in this article has been fully implemented by the author as a program written in the ``C'' programming language.
We show that the virtual Euler characteristics of the moduli spaces of $s$-pointed algebraic curves of genus $g$ can be determined from a polynomial in $1/γ$ where $γ$ permits specialization, through $γ=1,$ to the complex case treated by Harer and Zagier and, through $γ=1/2$, to the real case. This polynomial appears to have geometric significance, and may be the virtual Euler characteristic of some moduli space, as yet unidentified. This is related to a conjecture that the indeterminate $b=γ^1-1$ is associated with a combinatorial invariant of cell-decompositions through matrix models and the…
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric …
