Related papers: Hilbert scheme strata defined by bounding cohomolo…

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial…

We propose a general study of standard bases of polynomial ideals with parameters in the case where the monomial order is arbitrary. We give an application to the computation of the stratification by the local Hilbert-Samuel function.…

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $S$ be a smooth projective surface over $k$ with $k$-rational point $x$. For $n \geq 2$, let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. In this note, we…

In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally…

The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction,…

In order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it…

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial…

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $\omega$ and a projection onto $\mathbb{C}$. We provide a full…

Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the Hilbert series of binomial edge…

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family of…

We find an explicit formula for the elliptic stable envelope in the case of the Hilbert scheme of points on a complex plane. The formula has a structure of a sum over trees in Young diagrams. In the limit we obtain the formulas for the…

The study of the stratification associated to the number of generators of the ideals in the punctual Hilbert scheme of points on the affine plain goes back to the '70s. In this paper, we present an elegant formula for the E-polynomials of…

Given a homogeneous ideal I of a polynomial ring A=K[X_1,...,X_n] and a monomial order, we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to the order and denote it with…

We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the…

The NumericalHilbert package for Macaulay2 includes algorithms for computing local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. These techniques are numerically stable, and can be used…

Pursuing ideas of Jeff Smith, we develop a homotopy theory of ideals of monoids in a symmetric monoidal model category. This includes Smith ideals of structured ring spectra and of differential graded algebras. Such Smith ideals are NOT…

Let $\mathbf{k}$ be a closed field of characteristic zero. We prove that all monomial ideals sit in the curvilinear component of the Hilbert scheme of points of the affine space $\mathbb{A}_{\mathbf{k}}^n$, answering a long-standing…

Several moduli spaces parametrizing linear subspaces of the projective space are cut out by linear and quadratic equations in their natural embedding: Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper is to prove…

Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a…

Over an infinite field $K$ with $\mathrm{char}(K)\neq 2,3$, we investigate smoothable Gorenstein $K$-points in a punctual Hilbert scheme from a new point of view, which is based on properties of double-generic initial ideals and of marked…