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In this paper I present a new geometric approach to the factorization rulefor 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 itsnormalization is a locally trivial fibration over the moduli stack $VB(Y)$ ofvector 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 spaceof global sections of a power of the theta line bundle on $GVB(X)$ where thesummands are spaces of global sections of certain line bundles on the modulistack of parabolic bundles on the two-pointed curve $Y$.

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