Related papers: Equations for Chow and Hilbert Quotients

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that…

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family of…

The Witt group of a smooth curve over a real closed field is explicitely calculated. The method uses a comparison theorem between the graded Witt group and the etale cohomology groups. In the second part of the paper, the torsion Picard…

We study the locus of fixed points of a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the…

This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional…

In this paper we show that quotients of smooth projective toric varieties by $\mu_p$ in positive characteristics $p$ are toric varieties.

Let $\mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $\mathbb{P}^R$.…

We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus with respect to polarization spanned by almost-Hamiltonian vector fields of angle variables. The…

We prove arithmetic Riemann-Roch and Hilbert-Samuel type formulae for pointed stable curves. We give applications to volumes of lattices of integral cusps forms for pointed stable curves of genus 0.

We construct the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g…

In this expository article, we explain how to use localization to compute Gromov-Witten invariants of smooth toric varieties and orbifold Gromov-Witten invariants of smooth toric Deligne-Mumford stacks.

Wall-crossing formulas for various flavors of elliptic genus can be obtained using master spaces. We give a topological criterion which implies that such wall-crossing formulas are trivial. Applications are given for: GIT quotients,…

We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.

In this paper we study the Chow quotient ${\mathcal C}X$ of a convex variety $X$ of Picard number one by the action of a one dimensional torus having no non-trivial finite isotropy. Examples of these actions can be found in the rational…

Let $C$ be a smooth projective curve defined over the field of complex numbers. Let $E$ be a vector bundle on $C$, and fix an integer $d\geqslant 1$. Let $\mc Q:={\rm Quot}(E,d)$ be the Quot Scheme which parameterizes all torsion quotients…

We prove a non abelian Torelli type result for smooth projective curves by working in the derived category of some associated polarized Quot schemes and defining Brill-Noether loci and Abel-Jacobi maps on them.

In this paper we build bridges between moduli theory of sheaf stable pairs on one hand and birational geometry on the other hand. We will in particular treat moduli of sheaf stable pairs on smooth projective curves in detail and present…

For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal…

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of…

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…