Invariant vector bundles of rank 2 on hyperelliptic curves. (Q5954549)

The author studies rank two vector bundles on a hyperelliptic curve \(C\), with fixed determinant \(K_C\) and invariant under the hyperelliptic involution. Their locus \(\mathbf S^{\text{inv}}\subset \text{SU}_C(2,K_C)\) in the moduli space of semistable bundles turns out to have interesting geometric properties, generalizing the geometry of the Segre cubic threefold \(\Sigma_2=(\mathbb{P}^1)^6/\!/\text{PGL}(2,\mathbb{C})\). Namely, if one fixesNEWLINEa Weierstrass point \(w_0\) of \(C\), extensions of type \(0\to\mathcal O(-w_0)\to E\to K(w_0)\to 0\) induce a rational map \(\varepsilon : \mathbb{P}=NEWLINE\mathbb{P}(H^1(C,K_C^{-1}\otimes h^{-1}))\dashrightarrow \text{SU}_C(2,K_C)\) (here, \(h\) is the hyperelliptic line bundle). When \(\varepsilon\) is restricted to the invariant locus \(\mathbb{P}^+\subset \mathbb{P}\), it yields a rational map \(\varepsilon: \mathbb{P}^+ \dashrightarrow\mathbf S^{\text{inv}}\), whichNEWLINEis shown to be generically \(2:1\) onto its image \(S^i\) (this is a connected component of \(\mathbf S^{\text{inv}}\)). The branch locus turns out to be the Kummer variety of \(C\).NEWLINENEWLINEThis analogy to the genus 2 case motivates the definition of a generalized Segre variety \(\Sigma^g\) as the GIT quotient \((\mathbb{P}^1)^{2g+2}/\!/\text{PGL}(2,\mathbb{C})\), for the diagonal action and the natural linearization. Then, a general point of \(\Sigma_g\) represents a hyperelliptic curve of genus \(g\) with a special level-2 structure. The author proves that \(\Sigma_g\) can be obtained by the linear system of \(g\)-forms on \(\mathbb{P}^{2g-1}\), vanishing with multiplicity \(g-1\) through \(2g+1\) points in general position, generalizing the classical correspondence for the Segre cubic \(\sigma_2\). As an application to the genus 2 case, the author recovers a theorem of \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math. (2) 89, 14--51 (1969; Zbl 0186.54902)], stating \(\text{SU}_C(2,K_C)\simeq \mathbb{P}^3\).