A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles

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 $Y$.