Movable Curves and Semistable Sheaves

Linear Series on Semistable Curves

We study h 0 (X, L) for line bundles L on a semistable curve X of genus g, parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve and d = 0 or d = 2g − 2; X is stable, free from separating nodes, and d ≤ 4. These results are shown to be sharp. Applic...

Equivariant Versal Deformations of Semistable Curves

We prove that given any n-pointed prestable curve C of genus g with linearly reductive automorphism group Aut(C), there exists an Aut(C)equivariant miniversal deformation of C over an affine variety W . In other words, we prove that the algebraic stack Mg,n parametrizing n-pointed prestable curves of genus g has an étale neighborhood of [C] isomorphic to the quotient stack [W/Aut(C)].

Sheaves on Curves and Their Local Terms

Root Numbers of Semistable Elliptic Curves in Division Towers

The growth of the Mordell-Weil rank of an elliptic curve in a tower of number fields can be discussed at many levels. On the one hand, the issues raised can be embedded in a broader Iwasawa theory of elliptic curves (Mazur [10]); on the other hand, they can be crystallized in a single easily stated question, namely whether the rank of the elliptic curve over subextensions of finite degree is bo...

On Representations Attached to Semistable Vector Bundles on Mumford Curves

Introduction 1 1. Preliminaries 4 1.1. Notations and conventions 4 1.2. Reduction and formal rigid spaces 5 1.3. GAGA-results 8 1.4. Fundamental groups of analytic manifolds 14 1.5. Mumford curves 20 2. Representations attached to vector bundles 23 2.1. A review of the constructions 23 2.2. Comparison of the constructions 31 3. Applications 38 3.1. Fundamental groups of a Tate curve 39 3.2. Vec...