Related papers: Local BPS Invariants: Enumerative Aspects and Wall…

We consider local invariants of general connections (with torsion). The group of origin-preserving diffeomorphisms acts on a space of jets of general connections. Dimensions of moduli spaces of generic connections are calculated. Poincar\'e…

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…

The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2,R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of…

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are…

In the first part of this paper we provide a survey of some fundamental results about moduli spaces of framed sheaves on smooth projective surfaces. In particular, we outline a result by Bruzzo and Markushevich, and discuss a few…

As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic…

We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge…

We prove wall-crossing formulas for the motivic invariants of the moduli spaces of framed objects in the ind-constructible abelian categories. Developed techniques are applied in the case of the motivic Donaldson-Thomas invariants of…

Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times…

We prove that, for a natural class of Bridgeland stability conditions on $\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to…

The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on S^n bundles with nonnegative…

Maximal orders of rank 4 on the projective plane, ramified on a smooth plane quartic are examples of exotic del Pezzo orders. We compute the possible Chern classes for line bundles on such orders and show the moduli space of line bundles…

We apply virtual localization to the problem of finding blowup formulae for virtual sheaf-theoretic invariants on a smooth projective surface. This leads to a general procedure that can be used to express virtual enumerative invariants on…

In this paper, we study the Picard group of the moduli space of semistable sheaves on a smooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class. We focus especially on moduli spaces of sheaves of…

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for a general K3 surface…

Let $C$ be a smooth curve embedded in a smooth quasi-projective threefold $Y$, and let $Q^n_C=\textrm{Quot}_n(\mathscr I_C)$ be the Quot scheme of length $n$ quotients of its ideal sheaf. We show the identity…

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most basic intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth…

We consider the moduli space of stable vector bundles on curves embedded in P_2 with Hilbert polynomial 3m+1 and construct a compactification of this space by vector bundles. The result is a blow up of the Simpson moduli space…

A conjectural recursive relation for the Poincar\'e polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also…

We describe the invariants of plane quartic curves -- nonhyperelliptic genus 3 curves in their canonical model -- as determined by Dixmier and Ohno, with application to the classification of curves with given structure. In particular, we…