Related papers: Equations Defining Toric Varieties
Let X be a smooth simplicial toric variety. Let Z be the set of T-fixed points of X. We construct a filtration for A(Z), the ring of complex-valued functions on Z, such that Gr A(Z) is isomorphic to the cohomology algebra of X. This is the…
Let $X$ be a variety over a complete nontrivially valued field $K$. We construct an algebraizable formal model for the analytification of $X$ in the case $X$ admits a closed embedding into a toric variety. By algebraizable we mean that the…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals which generalize the hull complex of Bayer…
We consider the problem of deciding whether the solution sets of a parametrized polynomial system are toric in the sense that they admit a monomial parametrization. We focus on vertically parametrized systems, which are sparse systems where…
In this paper, we study toric ideals generated by circuits. For toric ideals which have squarefree quadratic initial ideals, a sufficient condition to be generated by circuits is given. In particular, squarefree Veronese subrings, the…
This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let $X$ be an…
Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics…
Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a…
These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected…
We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the…
Given a matrix Schubert variety $\overline{X_\pi}$, it can be written as $\overline{X_\pi}=Y_\pi\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_{\pi}$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action)…
This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable…
Multiplier ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety. More recently, they…
Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…
We provide a algebro-geometric combinatorial description of geometrically integral geometrically normal affine varieties endowed with an effective action of an algebraic torus over arbitrary fields. This description is achieved in terms of…
We examine Li's double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double…
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a…
We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical…
In this paper we give a toric representation of the associated ring of a polyomino which is obtained by removing a convex polyomino from its ambient rectangle.