Locus of non-very stable bundles and its geometry (Q1676390)

The author studies very stable rank 2 vector bundles on a complex projective curve of genus \(g \geq 2\), with special attention given to \(g=2\). A vector bundle \(E\) is said to be very stable if the only nilpotent Higgs field \(\phi : E \to E \otimes \Omega\) is zero. These vector bundles were defined by \textit{G. Laumon} [Duke Math. J. 57, No. 2, 647--671 (1988; Zbl 0688.14023)]. He proved that the very stable locus is a dense open in the moduli stack of all fixed degree vector bundles and remarked that Drinfeld had announced that this locus is, in fact, the complement of a pure codimension 1 substack. A proof of this last fact is not available in print, but the present paper proves the analogous statement (Theorem 1.1) for the moduli space of stable vector with rank \(2\) and fixed determinant of degree \(1\). When \(g=2\), this moduli space was explicitly described in terms of the classical geometry of quadratic hypersurfaces in [\textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. of Math. 89, No. 2, 14--51 (1969; Zbl 0186.54902)] and [\textit{U. V. Desale} and \textit{S. Ramanan}, Invent. Math. 38, No. 2, 161--185 (1976; Zbl 0323.14012)], and using that description, the author describes the complement of the very stable locus in an explicit, concrete fashion, showing e.g. that it is a surface of degree 32 (Theorem 1.3).