Varieties of Completely Decomposable Forms and Their Secants

Completely Decomposable Jacobian Varieties in New Genera

We present a new technique to study Jacobian variety decompositions using subgroups of the automorphism group of the curve and the corresponding intermediate covers. In particular, this new method allows us to produce many new examples of genera for which there is a curve with completely decomposable Jacobian. These examples greatly extend the list given by Ekedahl and Serre of genera containin...

On Spinor Varieties and Their Secants⋆

We study the secant variety of the spinor variety, focusing on its equations of degree three and four. We show that in type Dn, cubic equations exist if and only if n ≥ 9. In general the ideal has generators in degrees at least three and four. Finally we observe that the other Freudenthal varieties exhibit strikingly similar behaviors.

ar X iv : m at h / 05 03 34 0 v 1 [ m at h . A G ] 1 6 M ar 2 00 5 WEYL GROUPS AND ABELIAN VARIETIES

Let G be a finite group. For each integral representation ρ of G we consider ρ−decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with ρ(G)−action, of dimension equal to the degree of ρ, which admit a decomposition of the lattice for X into two G−invariant sublattices isotropic with respect to IH , with one of the sublattices ZG−isomorph...

Secants of Abelian Varieties, Theta Functions, and Soliton Equations

Line Arrangements Modeling Curves of High Degree: Equations, Syzygies and Secants

We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called graph curves can be embedded in projective space as line arrangements. We discuss property Np for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions reg...