On Newstead's conjecture on vector bundles on algebraic curves (Q1340172)

Let \(X\) be a smooth, complete curve of genus \(g \geq 2\) over an algebraically closed field \(k\) of characteristic 0. We write \(S_ L^{n,d}\) for the moduli variety of stable vector bundles of rank \(n\), degree \(d\) with determinant \(L\), \(L \in \text{Pic}^ d(X)\). The main result is the following: Theorem. Let \(n,d,r\) be natural numbers such that \(d = n(g - 1) + r\), \(0 < r < n\). Then there exists a non-empty Zariski-open subset of the moduli space \(S_ L^{n,d}\) consisting of vector bundles \(F\) such that (1) \(H^ 1(X,F) = 0\), (2) \({\mathcal O}^ r_ X\) is a subbundle of \(F\), (3) the quotient bundle \(F/{\mathcal O}^ r_ X\) is stable, where \({\mathcal O}^ r_ X\) is a trivial bundle of rank \(r\). This means that almost every vector bundle \(F\) can be represented as a vector bundle extension of the form \(0 \to {\mathcal O}^ r_ X \to E \to F \to 0\) (or its suitable twist). The above statement was conjectured by \textit{P. Newstead} [Math. Ann. 215, 251-268 (1975; Zbl 0288.14003); p. 255]; the theorem gives an elegant description of \(S_ L^{n,d}\) as a rational fibration over \(S_ L^{n-r,d}\). As a consequence we also prove the following: Let \(n,d\) be any natural numbers. Then there exists a non-empty Zariski-open subset of the moduli space \(S^{n,d}\) consisting of vector bundles \(F\) with natural cohomology (i.e., at most one cohomology group is nonzero).