Related papers: Computing Quot schemes via marked bases over quasi…

We describe three algorithms to determine the stable, semistable, and torus-polystable loci of the GIT quotient of a projective variety by a reductive group. The algorithms are efficient when the group is semisimple. By using an…

We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups $Ab$, and whose source category is an arbitrary category $\C$ with null object such that all objects are colimits of copies of a…

We build an infinite dimensional scheme parametrizing isomorphism classes of coherent quotients of a quasi-coherent sheaf on a projective scheme. The main tool to achieve the construction is a version of Grothendieck's Grassmannian…

Gr\"{o}bner bases are nowadays central tools for solving various problems in commutative algebra and algebraic geometry. A typical use of Gr\"{o}bner bases is the multivariate polynomial system solving, which enables us to construct…

We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts,…

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for…

This paper introduces the notion of an excellent quotient, which is stronger than a universal geometric quotient. The main result is that for an action of a connected solvable group $G$ on an affine scheme Spec$(R)$ there exists a…

We present basic properties of Gr\"obner bases of submodules of a free module of finite rank over a polynomial ring $R$ with coefficients in a graded truncated discrete valuations ring $A$. As an application, we give a criterion for a…

Let $X$ be a quasi projective scheme over a noetherian affine scheme $Spec(A)$, $U\subseteq X$ be an open subset, and $Z=X-U$.Assume that $Z$ is complete intersection, with $k=codim Z$. Consider the map $$ q:{\mathbb K}\left({\mathscr…

In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the…

In this paper we develop a Grobner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal…

G\"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes of points on a smooth algebraic surface and the Hodge numbers of generalized Kummer varieties. When a smooth projective surface $S$ admits an action by a finite…

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable…

For a semi-stable abelian variety A_K over a complete discrete valuation field K, we show that every finite subgroup scheme of A_K extends to a log finite flat group scheme over the valuation ring of K endowed with the canonical log…

Border basis schemes are open subschemes of the Hilbert scheme of $\mu$ points in an affine space $\mathbb{A}^n$. They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space…

For a given monomial ideal $J \subset k[x_1, \ldots, x_n]$ and a given monomial order $\prec$, the moduli functor of all reduced Gr\"obner bases with respect to $\prec$ whose initial ideal is $J$ is determined. In some cases, such a functor…

Suppose $S$ is an affine, noetherian scheme, $X$ is a separated, noetherian $S$-scheme, $\mathcal{E}$ is a coherent ${\mathcal{O}}_{X}$-bimodule and $\mathcal{I} \subset T(\mathcal{E})$ is a graded ideal. We study the geometry of the…

This paper is the continuation of \cite{CXY}. Let ${\bf G}$ be a simply connected semisimple algebraic group over $\Bbbk=\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), and $F$ be…

Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \ldots, d_n)$ w.r.t a system of weights $(w_{1},\dots,w_{n})$. Such systems are…

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over $\Bbbk$. Let $B$ be a connected projective scheme over $\Bbbk$ and let $\mathcal L$ be an ample…