A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles
Abstract
Description
In this paper I present a new geometric approach to the factorization rule for generalised theta functions.
Let $X$ be an irreducible projective nodal curve with one singularity and let $Y$ be its normalization. Recently I have constructed the moduli stack $GVB(X)$ of rank $n$ Gieseker vector bundles on $X$ and have shown that its normalization is a locally trivial fibration over the moduli stack $VB(Y)$ of vector bundles on $Y$, where the fibre is a canonical compactification of $Gl_n$. In this paper I prove a canonical direct sum decomposition of the space of global sections of a power of the theta line bundle on $GVB(X)$ where the summands are spaces of global sections of certain line bundles on the moduli stack of parabolic bundles on the two-pointed curve $Y$.
34 pages. The old version has been thoroughly revised in order to enhance readability. In a new chapter I prove by explicit dimension count that the space of generalized theta functions behaves well under degeneration
34 pages. The old version has been thoroughly revised in order to enhance readability. In a new chapter I prove by explicit dimension count that the space of generalized theta functions behaves well under degeneration