Let $k$ be an algebraically closed field of characteristic zero. Let $f:X\to S$ be a flat, projective morphism of $k$-schemes of finite type with integral geometric fibers. We prove the existence of a projective relative moduli space for semistable singular principal bundles on the fibers of $f$.

This generalizes the result of A. Schmitt, who studied the case when $X$ is a nodal curve.

Let $C$ be a smooth irreducible projective algebraic curve defined over the complex numbers. The notion of the Clifford index of $C$ was extended a few years ago to[4] semistable bundles of any rank. Recent work has been focused mainly on the rank-$2$ Clifford index, although interesting results have also been obtained for the case of rank $3$. In this paper we extend this work, obtaining improved lower bounds for the rank-$3$ Clifford index. This allows the first computations of the rank-$3$ index in nontrivial cases and examples for which the rank-$3$ index is greater than the rank-$2$ index.

We study the moduli stack of oriented orthogonal sheaves on a nodal curve and prove a factorization theorem.

In this paper we consider the iterated $G$-equivariant Hilbert scheme $G/N$-Hilb($N$-Hilb) and prove that $G/N\text{-}\mathrm{Hilb}\left(N\text{-}\mathrm{Hilb}\right({\mathbb{C}}^3\left)\right)$ is a crepant resolution of ${\mathbb{C}}^3/G$ isomorphic to the moduli space ${\mathcal{M}}_{\theta }\left(Q\right)$ of $\theta$-stable representations of the McKay quiver $Q$ for certain stability condition $\theta$. We provide several explicit examples to illustrate this construction. We also consider the problem of when $G/N$-Hilb($N$-Hilb) is isomorphic to $G$-Hilb showing the fact that these spaces are most of the times different.

We give sharp bounds on the vanishing of the cohomology of a tensor product of vector bundles on ${\mathbb{P}}^n$ in terms of the vanishing of the cohomology of the factors. For this purpose we introduce regularity indices generalizing the Castelnuovo–Mumford regularity.

As an application we give a sufficient condition for a vector bundle to have an unobstructed deformation theory that depends only on the cohomology table of the bundle. We construct complete families of bundles with such cohomology tables.

We shall show that the Kodaira dimension of the moduli scheme of stable sheaves on any Enriques surface is zero when its expected dimension is $7$ or more. A similar result holds also for bielliptic surfaces. We prove this by estimating the cup-product structure of ${Ext}^1$.

In this paper we continue our study of the moduli space of stable bundles of rank two and degree $1$ on a very general quintic surface. The goal in this paper is to understand the irreducible components of the moduli space in the first case in the “good” range, which is $c_2=10$. We show that there is a single irreducible component of bundles which have seminatural cohomology, and we conjecture that this is the only component for all stable bundles.

In this paper, we would like to propose a fundamental question about a higher-dimensional analogue of Dirichlet’s unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.