Related papers: Smooth Hilbert schemes: their classification and g…

Given a smooth curve on a smooth surface, the Hilbert scheme of the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural…

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal…

We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the…

Given an algorithm of resolution of singularities satisfying certain conditions (``good algorithms''), natural notions of simultaneous algorithmic resolution, or equiresolution, for families of embedded schemes (parametrized by a reduced…

Given a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider framed sheaves on $\mathscr X$, i.e., pairs $(\mathcal E,\phi_{\mathcal E})$, where $\mathcal E$ is a coherent…

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert…

We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded…

In the present paper we prove that the Hilbert scheme of 0-dimensional subspaces on supercurves of dimension $(1|1)$ exists and it is smooth. We show that the Hilbert scheme is not split in general.

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…

We discuss two conjectures by Francesco Severi and Joe Harris about the irreducibility and the dimension of the Hilbert scheme parameterizing smooth projective curves of given degree and genus.

We classify thick subcategories $\mathcal T \subset D^b(\mathrm{coh}\,C)$ for smooth projective curves $C$ over an algebraically closed field.

The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of…

Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.

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…

We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.

Given a singular projective variety in some projective space, we characterize the smooth curves contracted by the Gauss map in terms of normal bundles. As a consequence, we show that if the variety is normal, then a contracted line always…

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show…

We give a classification and a construction of all smooth $(n-1)$-dimensional varieties of lines in ${\bf P}\sp n$ verifying that all their lines meet a curve. This also gives a complete classification of $(n-1)$-scrolls over a curve…

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the…

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this…