Related papers: Two point extremal Gromov-Witten invariants of Hil…

In this paper, we study the Gromov-Witten theory of the Hilbert schemes X^{[n]} of points on smooth projective surfaces X with positive geometric genus p_g. Using cosection localization technique due to Y. Kiem and J. Li [KL1, KL2], we…

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the…

The cohomology of the Hilbert schemes of points on smooth projective surfaces can be approached both with vertex algebra tools and equivariant tools. Using the first tool, we study the existence and the structure of universal formulas for…

In This paper, we survey recent progress on the theory of Gromov- Witten invariants on Hilbert schemes of points mainly on elliptic surfaces and simply connected minimal surface of general type. In particular, we focus on the aspects of…

We show that a holomorphic two-form $\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\mgn(X,\beta)$ to the locus where $\theta$ degenerates; it then enables us to define the…

Using obstruction bundles, composition law and the localization formula, we compute certain 3-point genus-0 Gromov-Witten invariants of the Hilbert scheme of 3-points on the complex projective plane. Our results partially verify Ruan's…

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten…

In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme.…

We study automorphisms of the Hilbert scheme of $n$ points on a generic projective K3 surface $S$, for any $n \geq 2$. We show that the automorphism group of $S^{[n]}$ is either trivial or generated by a non-symplectic involution and we…

We analyse the geometry of Hilbert schemes of points on abelian surfaces and Beauville's generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of…

We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of $n$ points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus $0$ and for at…

Let $S$ be the affine plane regarded as a toric variety with an action of the 2-dimensional torus $T$. We study the equivariant Chow ring $A_{K}^*(Hilb^n(S))$ of the punctual Hilbert scheme $Hilb^n(S)$ with equivariant coefficients…

Let X be the quasi-projective symplectic surface that is given by the total space of the invertible sheaf O(-2) over the projective line. Let Hilb X be the family of Hilbert schemes of points on X. We give and prove a closed formula…

Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We…

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points.…

We prove that the Hilbert scheme of points on a normal quasi-projective surface with at worst rational double point singularities is irreducible.

We recursively compute the Gromov-Witten invariants of the Hilbert scheme of two points in the plane. By studying the space of stable maps and computing virtual contributions, we use these invariants to enumerate hyperelliptic plane curves…

We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification…

Various equivariant intersection numbers on Hilbert schemes of points on the affine plane are computed, some of which are organized into tau-functions of 2-Toda hierarchies. A correspondence between the equivariant intersection on Hilbert…

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…