Factorization of generalized Theta functions revisited

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.