Related papers: Smooth and irreducible multigraded Hilbert schemes

Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$ and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing effective…

We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions…

Let $R$ be the polynomial ring $K[x_{i,j}]$ where $1 \le i \le r$ and $j \in \mathbb{N}$, and let $I$ be an ideal of $R$ stable under the natural action of the infinite symmetric group $S_{\infty}$. Nagel--R\"omer recently defined a Hilbert…

Generalizing homogeneous spectra for rings graded by natural numbers, we introduce multihomogeneous spectra for rings graded by abelian groups. Such homogeneous spectra have the same completeness properties as their classical counterparts,…

We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture…

We describe the indecomposable components of the tangent bundle of the punctual Hilbert scheme of a smooth projective surface. As an application, we prove a recent conjecture about classification of products of punctual Hilbert schemes of…

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme…

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which…

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and…

We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions.…

Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function…

The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary…

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…

In this paper we give an effective characterization of Hilbert functions and polynomials of standard algebras over an Artinian equicharacteristic local ring; the cohomological properties of such algebras are also studied. We describe…

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\ldots,x^{p_t}y^{s_t},y^b] \subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion…

The Hilbert scheme of $n$ points in the affine plane contains the open subscheme parametrizing $n$ distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension $n$ supported at the origin of the affine…

Let $S$ be a smooth projective surface over $\mathbb{C}$ and $S^{[n]}$ be the Hilbert scheme of $n$ points over $S$, for any positive integer $n$. Let ${\bf a}=(n_1,\ldots,n_r)$ and ${\bf b}=(m_1,\ldots,m_s)$ be two distinct partitions of…

Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical…

It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting…

Let $K$ be a number field, let $X$ be a smooth integral variety over $K$, and assume that there exists a finite set of finite places $S$ of $K$ such that the $S$-integral points on $X$ are dense. Then the combined conjectures of Campana and…