Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on \(\mathbb P^1\) (Q2565944)

Let \(I\) denote a finite set of closed points on \({\mathbb P}^1\). Denote by \(M({\mathbb P}^1, I)\) the moduli space of semistable parabolic vector bundles of rank \(2\) with trivial determinant on \({\mathbb P}^1\) with parabolic structure at \(x\in I\) with weights \((0,\frac{1}{2})\). For \(I_0:= \{x_1,\cdots,x_{2g+2}\}\), let \(C\) denote the hyperelliptic curve whose branch locus is \(I_0\) and \(c_j\) the point of \(C\) over \(x_j\). Let \(M(C/{\mathbb P}^1,I_0)\) be the moduli space of rank \(2\) semistable vector bundles with trivial determinant on \(C\) together with an involution (a lift of the hyperelliptic involution on \(C\)) of fixed local type. Assume that the eigenvalues of the involution on the fibres of bundles at \(c_j\) are \(1, -1\) for all \(j\). For \(I_1= I_0 -{x_i}\) for some \(i\), let \(M(C/{\mathbb P}^1, I_1)\) denote the moduli space of rank \(2\) semistable vector bundles with determinant \({\mathcal O}(c_{2g+2})\) together with an involution such that the eigenvalues of the involution on the fibres of bundles at \(c_j\) are \(1, -1\) for \(1\leq j\leq 2g+1\) and at \(c_{2g+2}\) the eigenvalues are \(1, 1\) if \(g\) is odd and \(-1,-1\) if \(g\) is even. A result of \textit{U. N. Bhosle} says that for \(I= I_0\) or \(I_1\), \(M({\mathbb P}^1, I) \cong M(C/{\mathbb P}^1,I)\) [Math. Ann. 267, 347--364 (1984; Zbl 0521.14009)]. Hence \[ H^0(M({\mathbb P}^1, I), K^{-1}_{M({\mathbb P}^1, I)}) \cong H^0(M(C/{\mathbb P}^1,I), K^{-1}_{M(C/{\mathbb P}^1,I)}), \] where \(K\) denotes the canonical bundle. The author finds an explicit basis of \(H^0(M(C/{\mathbb P}^1,I), K^{-1}_{M(C/{\mathbb P}^1,I)})\) consisting of effective anticanonical divisors. There is a forgetful morphism from \(M(C/{\mathbb P}^1,I)\) to the moduli space \(M(2)\) of rank two vector bundles with fixed determinant on \(C\). \textit{A. Beauville} [Bull. Soc. Math. Fr. 116, No.~4, 431--448 (1988; Zbl 0691.14016); ibid. 119, No.~3, 259--291 (1991; Zbl 0756.14017)] had determined explicit bases of \(H^0(M(2), K^{-1/2}_{M(2)})\) parametrized by even (or odd) theta characteristics. The author determines the pullbacks of those bases (under \(f\)) in terms of the bases given by him.