Factorization of Generalized Theta Functions at Reducible Case

One of the problems in algebraic geometry motivated by conformal field theory is to study the behaviour of moduli space of semistable parabolic bundles on curve and its generalized theta functions when the curve degenerates to a singular curve. Let X be a smooth projective curve of genus g, and UX be the moduli space of semistable parabolic bundles on X , one can define canonically an ample line bundle ΘUX (theta line bundle) on UX and the global sections H 0(ΘkUX ) are called generalized theta functions of order k. These definitions can be extended to the case of singular curve. Thus, when X degenerates to a singular curve X0, one may ask the question how to determine H0(ΘkUX0 ) by generalized theta functions associated with the normalization X̃0 of X0. The so called fusion rules suggest that when X0 is a nodal curve the space H 0(ΘkUX0 ) decomposes into a direct sum of spaces of generalized theta functions on moduli spaces of bundles over X̃0 with new parabolic structures at the preimages of nodes. These factorizations and Verlinde formula were treated by many mathematicans from various points of view. It is obviously beyond my ability to give a complete list of contributions. According to [Be], there are roughly two approachs: infinite and finite. I understand that those using stacks and loop groups are infinite approach, and working in the category of schemes of finite type is finite approach. Our approach here should be a finite one. When X0 is irreducible with one node, a factorization theorem was proved in [NR] for rank two and generalized to arbitrary rank in [Su]. By this factorization, one can principally reduce the computation of generalized theta functions to the case of genus zero with many parabolic points. In order to have an induction machinery for the number of parabolic points, one should also prove a factorization when X0 has rwo smooth irreducible components intersecting at a node x0. This was done for rank two in [DW1] and [DW2] by analytic method. In this paper, we adopt the approach of [NR] and [Su] to prove a factorization theorem for arbitrary rank in the reducible case. Let I = I1 ∪ I2 ⊂ X be a finite set of points and U I X the moduli space of semistable parabolic bundles with parabolic structures at points {x}x∈I . When X degenerates to X0 = X1 ∪X2 and points in Ij (j = 1, 2) degenerate to |Ij| points x ∈ Ij ⊂ Xj r {x0}, we have to construct a degeneration UX0 := U I1∪I2 X1∪X2 of U X and theta line bundle ΘUX0 on it. Fix a suitable ample line bundle O(1) on X0,