Related papers: Nested Hilbert schemes and the nested q,t-Catalan …

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points…

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…

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…

For each braid $\beta\in Br_n$ we construct a $2$-periodic complex $\mathbb{S}_\beta$ of quasi-coherent $\mathbb{C}^*\times \mathbb{C}^*$-equivariant sheaves on the non-commutative nested Hilbert scheme $Hilb_{1,n}^{free}$. We show that the…

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on…

We compute the generating series for the intersection pairings between the total Chern classes of the tangent bundles of the Hilbert schemes of points on a smooth projective surface and the Chern characters of tautological bundles over…

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory…

We consider general difference equations $u_{n+1} = F(u)_n$ for $n \in \mathbb{Z}$ on exponentially weighted $\ell_2$ spaces of two-sided Hilbert space valued sequences $u$ and discuss initial value problems. As an application of the…

In this paper, we give new explicit representations of the Hilbert scheme of $\mu$ points in $\PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous…

Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is…

We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their…

We give formulas for the extension groups between tautological sheaves and more general between tautological objects twisted by a determinant line bundle on the Hilbert scheme of points on a smooth quasi-projective surface. We do this using…

We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory,…

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of…

In a previous paper, a realization of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads was given. We build upon that result to construct ADHM data for the Hilbert scheme of points of the total space…

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 develop a formula for tautological integrals over geometric subsets of the Hilbert scheme of points on complex manifolds. As an illustration of the theory, we derive a new iterated residue formula for the number of nodal curves in…

Let $S$ be a smooth projective surface over $\mathbb{C}$. We study the local and global geometry of the nested Hilbert scheme of points $S^{[n,n+1,n+2]}$. In particular, we show that $S^{[n,n+1,n+2]}$ is an irreducible local complete…

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the…

In this article, we describe the irreducible components of the Hilbert scheme of $d$ points on $\mathbb{A}^n$ for $d=9,10$. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a…