Related papers: Some Singular Lex-Segments

Let $k$ be an algebraically closed field and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We prove that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$, we…

In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new…

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.

We develop algorithms to compute two versions of the motivic Hilbert zeta function for curve singularities: the classical version, applicable to singularities with a monomial valuation semigroup or to singular curves defined by…

The paper investigates the locus of non-simple principally polarised abelian $g$-folds. We show that the irreducible components of this locus are $\Is^g_{D}$, defined as the locus of principally polarised $g$-folds having an abelian…

We develop a new method to study intersection theory of the main component of the Hilbert scheme of points on complex manifolds. The main result is an iterated residue formula for tautological integrals. We formulate a Chern-Segre-type…

In this article, we study the decomposition into irreducible components of the fixed point locus under the action of $\Gamma$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ of the smooth Nakajima quiver variety of the Jordan quiver. The…

Let $K$ be an algebraically closed field of characteristic 0, and let $H$ denote the Hilbert scheme of $m$ points of affine n-space $A^n$. An elementary component $E$ of $H$ is an irreducible component such that every $K$-point $[I]$ in $E$…

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 prove that the Gorenstein locus of the Hilbert scheme of points on $\mathbb A^n$ is non-reduced for $n\geq 12$; we construct examples of non-reduced points that come from apolar algebras of the sum of general cubics. As a corollary, we…

We study Gibbs partitions that typically form a unique giant component. The remainder is shown to converge in total variation toward a Boltzmann-distributed limit structure. We demon- strate how this setting encompasses arbitrary weighted…

Soit $H_{d,g}$ le sch\'ema de Hilbert des courbes lisses et connexes de degr\'e $d$ et genre $g$ de l'espace projectif ${\bf P}^3$ sur un corps $k$ alg\'ebriquement clos de caract\'eristique nulle. Le but principal de cet article est…

In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…

We show that the arithmetically Cohen--Macaulay (ACM) curves of degree 4 and genus 0 in ${\bold P}^4$ form an irreducible subset of the Hilbert scheme. Using this, we show that the singular locus of the corresponding component of the…

We introduce a natural structure of a semigroup (isomorphic to a factorization semigroup of the unity in the symmetric group) on the set of irreducible components of Hurwitz space of marked degree $d$ coverings of $\mathbb P^1$ of fixed…

We study the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ whose complete and very ample hyperplane linear series…

We introduce and study configuration schemes, which are obtained by ``glueing'' usual schemes along closed embeddings. The category of coherent sheaves on a configuration scheme is investigated. Smooth configuration schemes provide…

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected…

Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb P^r$. A component…

In this paper we study Hitchin system on singular curves. Some examples of such system were first considered by N. Nekrasov (hep-th/9503157), but our methods are different. We consider the curves which can be obtained from the projective…