Maximal line subbundles of stable bundles of rank 2 over an algebraic curve (Q2641352)

Let \(E\) denote a vector bundle of rank 2 over a smooth projective curve of genus \(g\geq 2\) over the field of complex numbers. The Segre invariant of \(E\) is defined by \(s(E)=\min(\deg E-2\deg L)\), where the minimum is taken over all line subbundles of \(E\). According to a theorem of Corrado Segre, \(s(E)\leq g\) and for a general such bundle either \(s(E)= g\) or \(s(E)=g-1\). A line subbundle \(L\) of \(E\) is called a maximal subbundle if \(s(E)=\deg E-2\deg L\). The set \(W(E)\) of maximal subbundles of \(E\) admits a natural scheme structure as a determinental variety. The main result of the paper is the following theorem: If \(E\) is a general vector bundle of rank 2 with fixed determinant, then the set \(W(E)\) determines the bundle \(E\). The proof for \(s(E)=g\) uses the geometric description of \(W(E)\) given by the reviewer and \textit{M. S. Narasimhan} [Math. Ann. 266, 55--72 (1983; Zbl 0507.14005)] and the fact that a general bundle is very stable in the sense of Laumon. The proof for \(s(E)=g-1\) uses elementary transformations.