Extending Oda's Theorem to Curves with Ordinary Singularities

Abstract

We extend a result of Tadao Oda (The first de Rham cohomology group and Dieudonn\'{e} modules, Cor. 5.11) to the case of a reduced proper curve \(X\) over a perfect field of characteristic \(p\) with at worst ordinary multiple points as singularities. Our result identifies the Dieudonn\'{e} module of the finite \(k\)-group scheme \(\picard^{0}_{X/k}[p]\) with a cohomological construction on \(X\) generalizing de Rham cohomology. Much of this thesis is devoted to developing relevant background material and giving a detailed review of Oda's proof. Our method of proof uses Oda's theorem and is not a replacement for it. However, one of our central results is an alternate characterization of Oda's map.