Related papers: Gorenstein toric Schubert varieties in Grassmannia…

We extend the work of Pappas-Rapoport-Zhu on twisted affine Grassmannians to wildly ramified, quasi-split, and residually split groups, assuming the maximal torus is induced. This relies on the construction, inspired by Tits, of certain…

In this paper, we provide a combinatorial description of seminormal toric varieties. The corresponding combinatorial object is a fan equipped with a collection of groups assigned to each cone. This framework introduces a more general class…

Perepechko and Zaidenberg conjectured that the neutral component of the automorphism group of a rigid affine variety is a torus. We prove this conjecture for toric varieties and varieties with a torus action of complexity one. We also…

We classify the non-toric, $\mathbb Q$-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for…

For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau-Ginzburg models as Laurent…

We give a closed-form formula for the Hilbert function of the tangent cone at the identity of a Schubert variety X in the Grassmannian in both group theoretic and combinatorial terms. We also give a formula for the multiplicity of X at the…

Let G be a connected complex semi-simple group, B a Borel subgroup of G, and T a maximal torus in B. We construct a class of smooth T-stable subvarieties inside the flag variety G/B, each of which is an embedding of a product of projective…

The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a…

Restriction varieties in the orthogonal Grassmannian are subvarieties of $OG(k, n)$ defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular, Schubert…

We prove sign-alternation of the product structure constants in the basis dual to the basis consisting of the structure sheaves of Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the partial flag…

Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety…

In this paper, we introduce the notion of "extension" of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or…

A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schr\"{o}der type as a smooth toric variety associated with a polygon…

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…

The closure of a generic torus orbit in the flag variety $G/B$ of type $A_{n-1}$ is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety $X_w$ $(w\in…

We prove a criterion for the normality of Schubert varieties in twisted affine Grassmannians in terms of the order of the algebraic fundamental group of a certain Levi subgroup, in particular in small positive characteristic. As an…

Let $LG_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $\lambda=(\lambda_1,...,\lambda_k)$ with $\lambda_1\leq n$…

For a fan $\Delta$, we introduce Grothendieck weights as a ring of functions from $\Delta$ to $\mathbb{Z}$ that form a K-theoretic analogue of Minkowski weights and describe the operational $K$-theory of a complete toric variety. We give an…

For $1\le r\le n-1,$ let $G_{r,n}$ denote the Grassmannian parametrizing $r$-dimensional subspaces of $\mathbb{C}^{n}.$ Let $(r,n)=1.$ In this article we show that the GIT quotients of certain Richardson varieties in $G_{r,n}$ for the…