Rank 2 parabolic bundles and generalized theta functions (Q1127651)

Let \({\mathcal N}= {\mathcal N} (k,\vec d)\) be the moduli space of rank 2 semi-stable parabolic vector bundles with fixed trivial determinant over a smooth projective curve \(C\). We determine the Picard group of \({\mathcal N}\) and define Cartier divisors in the linear system \(| {\mathfrak L}^{\text{par}} |\), where \({\mathfrak L}^{\text{par}}\) is an ample determinant line bundle over \({\mathcal N}\). If \(k=2\) and \(d_i=1\), we show an isomorphism between the space of global sections \(H^0 ({\mathcal N}, {\mathfrak L}^{\text{par}})\) and a direct sum of spaces of theta functions of order 4 over the Jacobian of \(C\).