Related papers: $IT_0$ sheaves on compactified Jacobian

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we…

We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we…

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper…

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is…

A plane curve on a the projective space over a field of characteristic zero is free if its associated sheaf T of tangent vector fields tangent is a free module. Relatively few free curves are known. Here we prove that a divisor consisting…

We use a complex version of the celebrated Atiyah-Hitchin-Drinfeld-Manin matrix equations to construct admissible torsion-free sheaves on $\p^3$ and complex quantum instantons over our quantum Minkowski space-time. We identify the moduli…

We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their…

We prove that there exist some stacks, representable over the stack of stable curves, having the following universal property with respect to N\'eron models of Jacobians. For every one-parameter family of stable curves, with regular total…

Let $f\col\C\ra B$ be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Abel--N\'eron map having values in Esteves's fine compactified Jacobian and extending the degree-2 Abel--Jacobi map of the…

We show that certain ramification invariants associated to a compatible system of $\ell$-adic sheaves on a curve are independent of $\ell$.

The purpose of this paper is to study the cohomology rings of universal compactified Jacobians. Over the moduli space $\overline{\mathcal{M}}_{g,n}$ of Deligne-Mumford stable marked curves with $n\geq 1$, on the one hand we show that the…

We prove autoduality for curves of compact type and, more generally, treelike curves with planar singularities. More precisely, we produce an isomorphism between the generalized Jacobian of such a curve and the connected component of the…

We define the analogue of instanton sheaves on the blow-up $\widetilde{\mathbb{P}^n}$ of the $n-$dimensional projective space at a point. We choose appropriate polarisation on $\widetilde{\mathbb{P}^n}$ and construct rank $2$ examples of…

Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe…

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a…

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $\alpha_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In…

In this work, we present an efficient method for computing in the generalized Jacobian of special singular curves, nodal curves. The efficiency of the operation is due to the representation of an element in the Jacobian group by a single…

We construct a good compactification of the variety of irreducible projective plane curves of degree n with d nodes and no other singularities.

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli…

We completely describe all semi-stable torsion free sheaves of degree zero on nodal cubic curves using the technique of Fourier-Mukai transforms. The Fourier-Mukai images of such sheaves are torsion sheaves of finite length, which we…