Irregular and singular loci of commuting varieties

Cohomological support loci and Pluricanonical systems on irregular varieties

For an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic condition, the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ . We then show that the condition that the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.

Irregular locus of the commuting variety of reductive symmetric Lie algebras and rigid pairs

The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…

Critical loci for Higgs bundles

The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bundles. Topological and differential-geometric properties of the critical loci are addressed.

2-步理想与交换矩阵

2-step ideals and commuting matrices

本文利用 Hilbert 方案与交换矩阵的对应关系，定义与 2-步理想对应的矩阵轨迹，估计其维数，为 n=3 情形下 2-步理想轨迹维数提供新证明，并省略部分假设。

Brill-Noether loci for divisors on irregular varieties

For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={η\in Pic^0(X)|h^0(L+η)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number ρ(C, r) and we prove, under some mild additional assumptions, that if ρ(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to ρ(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly o…

Castelnuovo-Mumford Regularity and GV-Sheaves on Irregular Varieties

Inspired by Beauville's recent construction of Ulrich sheaves on abelian surfaces, we pose the question of whether a torsion-free sheaf on a polarized smooth projective variety with Castelnuovo-Mumford regularity 1 is a GV (generic vanishing) sheaf, and present evidence that this question is governed by the positivity of cycles on generalized Brill-Noether loci. We prove that it has an affirmative answer for natural polarizations on many well-known irregular surfaces, as well as some polarizations on ruled threefolds over a curve.