Related papers: An introduction to Hilbert schemes of points on AD…

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of…

An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams $E_6, E_7$, $E_8$. These curves are…

This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction…

Let $G$ be a finite subgroup of $\mbox{GL}(2)$ acting on $\mathbf{A}^2\setminus\{0\}$ freely. The $G$-orbit Hilbert scheme $G\mbox{-Hilb}(\mathbf{A}^2)$ is a minimal resolution of the quotient $\mathbf{A}^2/G$. We determine the generator…

The criterion for an affine primary algebra over the field to be integral, is proven. Using this criterion we give a simple proof that Hilbert scheme of 0-dimensional subschemes of length $l$ of nonsingular $d$-dimensional algebraic variety…

We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A-infinity-algebras. The tangent…

We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C^2)^6 as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure…

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical…

In this paper we prove two results which further classify smoothness properties of Hilbert schemes of points. This is done by counting classes of arrows on Young diagrams corresponding to monomial ideals, building on the approach taken by…

We relate Hilbert schemes of points and Fulton-MacPherson compactifications by an interpolating stability condition. We then derive wall-crossings formulas and some applications for the enumerative geometry of Hilbert schemes.

In this paper we construct $G$-Hilbert schemes for finite group schemes $G$. We find a construction of $G$-Hilbert schemes as relative $G$-Hilbert schemes over the quotient that does not need the Hilbert scheme of $n$ points, works under…

We combine our results on symmetric products and second quantization with our description of discrete torsion in order to explain the ring structure of the cohomology of the Hilbert scheme of points on a K3 surface. This is achieved in…

We give an explicit stratification of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^{m+1}$ with respect to $m$-dimensional partitions in the Grothendieck group of varieties. As an application, we calculate the classes of the…

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this…

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…

We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme…

We construct irrational irreducible components of the Hilbert scheme of points of affine n-dimensional space, for n at least 12. We start with irrational components of the Hilbert scheme of curves in P^3 and use methods developed by…

Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3,…

We present a method based on mutations of helices which leads to the construction (in the large volume limit) of exceptional coherent sheaves associated with the $(\sum_al_a=0)$ orbits in Gepner models. This is explicitly verified for a few…

We construct nonnatural automorphisms of the Hilbert scheme of two points of some simple abelian variety preserving the big diagonal by considering automorphisms of the n-th product of the abelian varieties.