Related papers: Matrix factorizations and intermediate Jacobians o…

We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve $X$, the relative smooth locus of a relative fine compactified Jacobian is…

In this paper we describe compactified universal Jacobians, i.e. compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to…

This paper studies the components of the moduli space of rank 1, torsion-free sheaves, or compactified Jacobian, of a non-Gorenstein curve. We exhibit a generically reduced component of dimension equal to the arithmetic genus and prove that…

A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1…

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set…

We use explicit blow-ups and computations of birational Fujita-Zariski decompositions to determine generic infinitesimal Newton-Okounkov bodies for box-product ample polarizations on three classes of spaces: product between a curve and the…

For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration…

A pair of disjoint lines on a smooth cubic threefold determines an irreducible component of the Hilbert scheme. We prove that this component is smooth and isomorphic to the blow-up of the symmetric product of Fano varieties of lines on the…

A basic technique for studying a family of Jacobian varieties is to extend the family by adding degenerate fibers. Constructing an extension requires a choice of fibers, and one typically chooses to include either degenerate group varieties…

The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing…

We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal…

This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are…

By the description due to Mukai and Iliev, a smooth prime Fano threefold X of genus 9 is associated to a surface P(V), ruled over a smooth plane quartic Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X with odd…

The paper studies representation theoretic aspects of a nonabelian version of the Jacobian for a smooth complex projective surface $X$ introduced in [R1]. The sheaf of reductive Lie algebras $\bf\calG$ associated to the nonabelian Jacobian…

We consider the moduli space of rigidified perfect complexes with support on a general complete intersection Calabi-Yau threefold $X$ and its Tyurin degeneration $X\rightsquigarrow X_1\cup_SX_2$ to a complete intersection of Fano threefolds…

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra…

This article generalizes the correspondence between matrix factorizations and maximal Cohen-Macaulay modules over hypersurface rings due to Eisenbud and Yoshino. We consider factorizations with several factors in a purely categorical…

Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study them in relation with the Cartan cubic, the $E_6$-invariant cubic in $\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a linear section of…

We continue our study of the geometry of Nieto's quintic threefold, looking at degenerate surfaces that correspond to certain loci and showing how they arise from a toroidal compactification of a suitable moduli space.

Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface $Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves contained in $X$…