Moduli of torsionfree sheaves of rank two and odd degree on a nodal hyperelliptic curve
Let \(X\) be an irreducible nodal hyperelliptic curve of arithmetic genus \(g\) defined over an algebraically closed field \(k\) of characteristic \(0\) and let \(L\) be a line bundle of odd degree \(d\) on \(X\). Let \(U_L'(2,d)\) be the moduli space of stable bundles of rank \(2\) and determinant \(L\) on \(X\) and let \(U_L(2,d)\) be the closure of \(U_L'(2,d)\) in the moduli space of torsion-free sheaves of rank \(2\) and degree \(d\). In this paper, the author obtains an explicit geometric description of \(U_L(2,d)\), which generalises that given by \textit{U. V. Desale} and \textit{S. Ramanan} for the case where \(X\) is smooth [Invent. Math. 38, 161--185 (1976; Zbl 0323.14012)]. If \(X\) has \(m\) nodes, it determines a pencil of quadrics in \({\mathbb P}^{2g+1}\) with Segre symbol \([2\cdots 2 1\cdots1\)] with \(2\) repeated \(m\) times and \(1\) repeated \(2g+2-2m\) times. Let \(R\) denote the scheme of \((g-2)\)-dimensional linear subspaces of \({\mathbb P}^{2g+1}\) which are isotropic for every member of the pencil. The scheme \(R\) is Cohen-Macaulay and normal (Proposition 3.7). In the case \(m=1\), let \(R_0\) be the subscheme of \(R\) consisting of those subspaces which contain the unique singular point of the intersection of the pencil of quadrics; then \(R':=R-R_0\) is smooth and irreducible (Corollary 3.6). In the case \(m=1\), the author proves in Theorem 1.2 that \(R\) is the normalisation of \(U_L(2,d)\). In fact there is a morphism \(f:R\to U_L(2,d)\) such that the restrictions of \(f\) to \(R'\) and \(R_0\) are isomorphisms onto \(U_L'(2,d)\) and \(U_L(2,d)-U_L'(2,d)\). For \(g=2\), \(R\) is an intersection of quadrics in \({\mathbb P}^5\) and \(R_0\) is the unique singular point of \(R\). Even in this case, it is not known whether \(U_L(2,d)\) is normal. For general \(m\), Theorem 1.3 states that there is a birational surjective morphism \(f:R\to U_L(n,d)\). In the smooth case, a crucial ingredient in the 1976 proof was the fact that, if \(E\) is a stable bundle of rank \(2\) on \(X\), then \(E\otimes i^*E\) is semistable, where \(i\) is the hyperelliptic involution. This made it possible to deduce the theorem from the corresponding theorem for orthogonal bundles of rank \(4\) with \(i\)-action. In the singular case, it is no longer true that \(E\otimes i^*E\) is semistable. However, this does hold on a non-empty open subset \(U\) and, outside \(U\), the author uses her concept of generalised parabolic bundles. This enables her, in the case \(m=1\), to use the methods of the paper cited above to construct a morphism \(f':R'\to U_L'(2,d)\) and an inverse for this morphism on \(U\). The morphism \(f'\) is then extended to \(f:R\to U_L(2,d)\) and the remaining properties stated in the theorem are proved. When \(m>1\), some part of this argument works and is supplemented by an induction on \(m\).
scientific article; zbMATH DE number 6146770
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