Factorization of generalized Theta functions revisited

TL;DR

This paper revisits the factorization of generalized Theta functions, providing new insights into moduli spaces, independence of dimensions, and canonical bundle computations, with a focus on reducible curves.

Contribution

It introduces new methods to analyze moduli spaces of generalized parabolic sheaves without relying on vanishing theorems, and offers novel calculations for reducible curves.

Findings

Dimension of global sections is independent of choices for smooth curves.

New estimates of codimension in moduli spaces.

Computed canonical line bundle for moduli of generalized parabolic sheaves on reducible curves.

Abstract

This survey is based on my lectures given in last a few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorem, we show, without using vanishing theorem, a new observation that ${\rm dim}\,H^0(\sU_C,\Theta_{\sU_C})$ is independent of all of the choices for any smooth curves. The estimate of various codimension and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.