Line bundles on the moduli space of parabolic connections over a compact Riemann surface

Let X be a compact Riemann surface of genus g >= 3 and S a finite subset of X. Let xi be fixed a holomorphic line bundle over X of degree d. Let M-pc(r, d, alpha) (respectively, M-pc(r, alpha, xi)) denote the moduli space of parabolic connections of rank r, degree d and full flag rational generic weight system alpha, (respectively, with the fixed determinant xi) singular over the parabolic points S subset of X. Let M-pc'(r, d, alpha) (respectively, M-pc'(r, alpha, xi)) be the Zariski dense open subset of M-pc(r, d, alpha) (respectively, M-pc(r, alpha, xi)) parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces M-pc'(r, d, alpha), and M-pc'(r, alpha, xi) by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces M-pc(r, d, alpha), and M-pc(r, alpha, xi). Let C(L) denote the space of holomorphic connections on an ample line bundle L over the moduli space M(r, d, alpha) of parabolic bundles. We show that C(L) does not admit any non-constant algebraic function. (C) 2022 Elsevier Inc. All rights reserved.