Searcharxiv⌕ Search

arXiv subjects

Valentina Kiritchenko

Publications and source records attributed to Valentina Kiritchenko.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial valuations on plane polygons

Scissors congruence problems involving translations have prompted the study of translation invariant simple valuations. We review this classical theory from a naive and consistent viewpoint: starting from a description of all simple valuations on polygons, we characterize the effect of translation invariance. A description of all polynomial simple valuations is obtained as a bi-product of the adopted approach and as a direct generalization of the translation invariant theory; it appears to be new.

math.MG↗
arXiv2026

Valuations on polyhedra and topological arrangements

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology.

math.CO↗
arXiv2024

Quadratic residue patterns, algebraic curves and a K3 surface

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive $\ell$-tuples of quadratic residues, studying corresponding algebraic curves and their jacobians, which happen to be products of jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.

math.AG↗
arXiv2023

Quadratic residue patterns and point counting on K3 surfaces

Quadratic residue patterns modulo a prime are studied since 19th century. We state the last unpublished result of Lydia Goncharova, reformulate it and prior results in terms of algebraic geometry, and prove it. The core of this theorem is an unexpected relation between the number of points on a K3 surface and that on a CM elliptic curve.

math.AG↗
arXiv2022

Simple geometric mitosis

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations give a uniform construction of Knutson-Miller mitosis (in type A) and (simplified) Fujita mitosis (in type C) on Kogan faces of Gelfand-Zetlin polytopes.

math.CO↗
arXiv2020

Push-pull operators on convex polytopes

A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their applications to the theory of Newton-Okounkov convex bodies. Convex geometric push-pull operators yield an inductive construction of Newton-Okounkov polytopes of Bott-Samelson varieties. In particular, we construct a Minkowski sum of Feigin-Fourier-Littelmann-Vinberg polytopes using convex geometric push-pull operators in type A.

math.AG↗
arXiv2019

Newton-Okounkov polytopes of flag varieties for classical groups

For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell and is combinatorially related to the Gelfand-Zetlin pattern in the same type. In types A and C, we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. In types B and D, we compute low-dimensional examples and formulate open questions.

math.AG↗
arXiv2018

Schubert calculus on Newton-Okounkov polytopes

A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the general framework and survey particular realizations of this approach in types A, B and C.

math.AG↗
arXiv2017

Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums

We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $GL_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition $(s_1)(s_2s_1)(s_3s_2s_1)(\ldots)(s_{n-1}\ldots s_1)$ of the longest element in the Weyl group, and the Schubert subvarieties correspond to the terminal subwords in this decomposition. We prove that the resulting Newton--Okounkov polytopes for semiample line bundles satisfy the additivity property with respect to the Minkowski sum. In particular, they are Minkowski sums of Newton--Okounkov polytopes of line bundles on the complete flag varieties for $GL_2$,\ldots, $GL_{n}$.

math.AG↗
arXiv2016

Newton-Okounkov polytopes of flag varieties

We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton--Okounkov bodies coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.

math.AG↗
arXiv2015

Merging divisorial with colored fans

Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans. The torus in question is obtained as the identity component of the quotient group N/H, where N is the normalizer of H in G. The resulting Chow quotient is equal to (a blowup of) the simple toroidal compactification of G/(H N^0). In the horospherical case, for example, it is equal to a flag variety, and the slices (coefficients) of the divisorial fan are merely shifts of the colored fan along the colors.

math.AG↗
arXiv2014

Geometric mitosis

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.

math.AG↗
arXiv2013

Divided difference operators on polytopes

We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductive groups. In particular, Gelfand-Zetlin polytopes and twisted cubes of Grossberg-Karshon are obtained in a uniform way.

math.AG↗
arXiv2012

Number of vertices in Gelfand-Zetlin polytopes

We discuss the problem of counting vertices in Gelfand-Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas.

math.CO↗
arXiv2011

Schubert calculus and Gelfand-Zetlin polytopes

We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.

math.AG↗
arXiv2011

Equivariant Cobordism of Flag Varieties and of Symmetric Varieties

We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational cobordism rings of wonderful symmetric varieties of minimal rank.

math.AG↗
arXiv2009

Gelfand-Zetlin polytopes and flag varieties

I construct a correspondence between the Schubert cycles on the variety of complete flags in C^n and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SL_n(C) with a strictly dominant highest weight. The construction is based on a geometric presentation of Schubert cells by Bernstein-Gelfand-Gelfand using Demazure modules. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand-Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.

math.AG↗