Derivation Algebras of Toric Varieties (original) (raw)

Lie algebras of vertical derivations on semiaffine varieties with torus actions

Journal of Pure and Applied Algebra, 2021

Let X be a normal variety endowed with an algebraic torus action. An additive group action α on X is called vertical if a general orbit of α is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of α in Aut(X). Our first result in this paper is a classification of vertical additive group actions on X under the assumption that X is proper over an affine variety. Then we establish a criterion as to when the infinitesimal generators of a finite collection of additive group actions on X generate a finite-dimensional Lie algebra inside the Lie algebra of derivations of X.

A note on affine toric varieties

Linear Algebra and Its Applications, 2000

Let k be an arbitrary field and a toric set in the affine space A n k given parametrically by monomials. Using linear algebra we give necessary and sufficient conditions for to be an affine toric variety, and show some applications.

Notes on toric varieties

These notes survey some basic results in toric varieties over a field F , with examples and applications.

Toric Varieties and Their Applications

Toric Varieties and Their Applications, 2022

The thesis provides an introduction into the theory of affine and abstract toric varieties. In the first chapter, tools from algebraic geometry indispensable for the comprehension of the topic are introduced. Many properties of convex polyhedral cones and affine toric varieties are proven and discussed in detail as is the deep connection between the two objects. The second chapter establishes the notion of an abstract variety and translates obtained results to this more general setting, giving birth to the theory of abstract toric varieties and the closely associated theory of fans. Finally, an algorithmic approach to the resolution of singularities on toric surfaces and its relation to continued fractions is revealed.

Toric varieties from cyclic matrix semigroups

2020

We present and expand some existing results on the Zariski closure of cyclic groups and semigroups of matrices. We show that, with the exclusion of isolated points, their irreducible components are toric varieties. Additionally, we demonstrate how every toric variety can be realized as the Zariski closure of a cyclic matrix group. Our paper includes a number of explicit examples and a note on existing computational results. Introduction In mathematics, as well as in many applied sciences, researchers often face the problem of describing a complicated behaviour or a sophisticated model. A common approach is to find invariants: roughly speaking, an invariant is a property shared by every point of the model or a function that attains the same value at every state. Invariants appear in a wide range of areas of mathematics, physics, and computer science. As an example, in the study of dynamical systems, invariants can determine whether the system will reach a given state. From an algebra...

Syzygies of affine toric varieties

Journal of Algebra, 2000

We give a method for computing the syzygies of the coordinate ring R of an affine toric variety. We show how the method works for dimension one and two cases, Cohen-Macaulay semigroups, and for computing minimal generators of the defining ideal. We show how to compute the depth of R and generalize a criterion for Cohen-Macaulayness.

Toric geometry and the Semple–Nash modification

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2012

This paper proposes some material towards a theory of general toric varieties without the assumption of normality. Their combinatorial description involves a fan to which is attached a set of semigroups subjected to gluing-up conditions. In particular it contains a combinatorial construction of the blowing up of a sheaf of monomial ideals on a toric variety. In the second part this is used to show that iterating the Semple-Nash modification or its characteristic-free avatar provides a local uniformization of any monomial valuation of maximal rank dominating a point of a toric variety.

Toric varieties from cyclic matrix groups

arXiv: Algebraic Geometry, 2020

We study cyclic groups and semigroups of matrices from an algebraic geometric viewpoint. We present and expand some existing results and highlight their connection with toric geometry.

Toda lattice and toric varieties for real split semisimple Lie algebras

1999

The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan subgroup of the corresponding Lie group tildeG\tilde GtildeG which is a disjoint union of the split Cartan subgroups associated to semisimple portions of Levi factors of all standard parabolic subgroups of tildeG\tilde GtildeG. The manifold is also related to the compactified level sets of a generalized Toda lattice equation defined on the semisimple Lie algebra, which is diffeomorphic to a toric variety in the flag manifold tildeG/B{\tilde G}/BtildeG/B with Borel subgroup BBB of tildeG\tilde GtildeG. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored-signed Dynkin diagrams which parametrize the cells in the decomposition.

Riemann-Roch for Quotients and Todd Classes of Simplicial Toric Varieties

Communications in Algebra, 2003

In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne [BV] using different techniques.