Moduli space of parabolic vector bundles over hyperelliptic curves
Let \(Y\) be an irreducible, smooth, projective curve over \(\mathbb{C}\) with finite automorphism group. Let \(\Gamma\) be a subgroup of \(\mathrm{Aut}(Y)\) and assume that the quotient \(X:= Y/\Gamma\) exists and is smooth. Let \(p: Y \to X\) be the quotient map, \(D\) be the ramification divisor, and \(p^{-1}D=\sum_{i=1}^r y_i\). Let \(\delta=\mathcal{O}_Y(\sum_{i=1}^r n_iy_i)\) where all the \(n_i\)'s are odd integers except one, say \(n_1\), and \(\mathcal{M}\) be the moduli space of rank 2 stable bundles on \(Y\) with fixed determinant \(\delta\). There is a natural action of \(\Gamma\) on \(\mathcal{M}\), and let \(\mathcal{M}^{\Gamma}\) denotes the \(\Gamma\)-fixed points of \(\mathcal{M}\). Let \(\mathcal{M}_P\) denotes the moduli space of rank 2 parabolic stable bundles with fixed determinant \(p_*^{\Gamma}\delta\) (the \(\Gamma\)-invariant subsheaf of \(p_* \delta\)) with parabolic divisor \(D\) and parabolic wights \((0,0)\) at \(p(y_1)\) and \((0,\frac{1}{2})\) at all other points. The authors showed that for \(\Gamma=\mathbb{Z}/2\mathbb{Z}, \mathcal{M}^{\Gamma} \simeq \mathcal{M}_P\). The authors then proceeded to show that given any smooth, projective irreducible hyperelliptic curve \(X\) of genus \(g\), one can construct a unique hyperelliptic curve \(Y\) of genus \(2g+1\) together with a morphism \(p:Y \to X\) such that \(Y\) is a two-sheeted cover of \(X\). When \(\delta= \mathcal{O}_Y(4gy_1+y_2+y_3+y_4)\), using the above observations the authors gave a geometric description of \(\mathcal{M}_P\) analogous to the one given in \textit{U. V. Desale} and \textit{S. Ramanan}, Invent. Math. 38, 161--185 (1976; Zbl 0323.14012).
scientific article; zbMATH DE number 6990975