SU(r,L) is separably unirational

On the Picard group of moduli spaces

We study the Picard groups of moduli spaces in positive characteristics and we give a "$p$-adic" proof that the Picard group of moduli of vector bundles of fixed determinant is isomorphic to the group of integers. Along the way we prove that the local fundamental group scheme of a normal unirational projective variety is trivial. This is reminiscent of results of Serre and Nygaard who studied the fundamental groups of smooth, projective, unirational varieties.

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds

We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^{-1}.

关于模空间中稳定束的Brill-Noether亚变种的小余维数注记

A note on the Brill-Noether loci of small codimension in moduli space of stable bundles

本文研究了稳定束模空间中Brill-Noether亚变种的小余维数性质，证明了不同亏格情况下该亚变种的理性质和Chow群平凡性。

Even sets of nodes on sextic surfaces

We determine the possible even sets of nodes on sextic surfaces in $\Pn 3$, showing in particular that their cardinalities are exactly the numbers in the set $\{24, 32, 40, 56 \}$. We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification of existence relies on computer calculations. The construction gives a maximal family…