Related papers: Toric matrix Schubert varieties and their polytope…

For each $A\in\N^n$ we define a Schubert variety $\sh_A$ as a closure of the $\Slt(\C[t])$-orbit in the projectivization of the fusion product $M^A$. We clarify the connection of the geometry of the Schubert varieties with an algebraic…

The purpose of this article is to extend the earliest results of A.A. Brudno, connecting topological entropy of a subshift X over $\mathbb{N}$ to the Kolmogorov complexity of words in X, to subshifts over computable groups that posses…

The theory of polyptych lattices is a framework to obtain a family of toric degenerations whose polytopes are related by piecewise-linear transformations. It can be regarded as a generalization of toric degenerations arising from cluster…

Using the method of degenerating a Grassmannian into a toric variety, we calculate recursive formulas for the dimensions of the eigenspaces of the action of an n-dimensional torus on a Grassmannian of planes in an n-dimensional space. In…

We study compactifications of Matrix theory on twisted tori and non-commutative versions of them. As a first step, we review the construction of multidimensional twisted tori realized as nilmanifolds based on certain nilpotent Lie algebras.…

These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any…

We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal…

We describe maximal commutative unipotent subgroups of the automorphism group $\mathrm{Aut}(X)$ of an irreducible affine variety $X$. Further we show that a group isomorphism $\mathrm{Aut}(X) \to \mathrm{Aut}(Y)$ maps unipotent elements to…

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…

We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization…

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L :=…

Following Mitchell's philosophy, in this paper we define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two additive categories $\mathcal{U}$ and $\mathcal{T}$ and $M\in…

For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$: we call these closures large Schubert varieties. We show that these…

An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work…

The Peterson variety is a subvariety of the flag manifold $G/B$ equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert…

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the…

The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "Schubert variety of hyperplane arrangement". In contrast, the lattice of flats of a…

The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over…

We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in…

Let $X$ be a projective variety with a torus action, which for simplicity we assume to have dimension 1. If $X$ is a smooth complex variety, then the geometric invariant theory quotient $X//G$ can be identifed with the symplectic reduction…