Related papers: Invariants of nested Hilbert and Quot schemes on s…

In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\TT^2$-equivariant locally free sheaf on…

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…

We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for $\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots of…

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating…

We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface $S \to \Sigma$ for all curve classes which are contracted by the induced fibration $S^{[n]} \to \Sigma^{[n]}$. The formula…

In the present paper, we show that the motivic Hilbert zeta function for a curve singularity yields the generating functions for Euler numbers of punctual Hilbert schemes when any punctual Hilbert scheme admits an affine cell decomposition.…

This paper presents new examples of elementary and non-elementary irreducible components of the Hilbert scheme of points and its nested variants. The results are achieved via a careful analysis of the deformations of a class of finite…

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…

Let $S$ be a smooth projective surface over $\mathbb{C}$. Let $S^{[n_1,\dots,n_k]}$ denote the nested Hilbert scheme which parametrizes zero-dimensional subschemes $\xi_{n_1} \subset \ldots \subset \xi_{n_k}$ where $\xi_i$ is a closed…

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface. Let $S$ be a K3 surface and let $\mathsf{Hilb}^d(S)$ be the Hilbert scheme of $d$ points of $S$. In case of elliptically fibered K3…

We propose a variation of the classical Hilbert scheme of points - the double nested Hilbert scheme of points - which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth…

We use basic algebraic topology and Ellingsrud-Stromme results on the Betti numbers of punctual Hilbert schemes of surfaces to compute a generating function for the Euler characteristic numbers of the Douady spaces of "n-points" associated…

In this note we investigate the generating series of the Euler characteristics of Hilbert scheme of points on cyclic quotient singularities of type (p,1). We link the appearing combinatorics to p-fountains, a generalization of the notion of…

Let $\Sigma_{g,r}$ denote the $r$-punctured closed Riemann surface of genus $g$. For every $g\geq 0$, we determine the four-variable generating function for the mixed Hodge numbers of the unordered configuration spaces of $\Sigma_{g,1}$.…

This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite…

Let $C$ be a smooth curve. In this paper we investigate the geometric properties of the double nested Hilbert scheme of points on $C$, a moduli space introduced by the third author in the context of BPS invariants of local curves and sheaf…

This paper delves into the study of Hilbert schemes of unibranch plane curves whose points have a fixed number of minimal generators. Building on the work of Oblomkov, Rasmussen and Shende we provide a formula for their motivic classes and…

Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The…

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for…