Families of maximal subbundles for rank two bundles on a curve (Q1359514)

For \(E\) a rank two vector bundle over a curve \(X\) of genus \(g\) define: \(s(E)= \deg(E)- 2\max\deg(L)\), taking the maximum over all line bundles \(L\subset E\). \textit{Nagata} showed \(s\leq g\), and \textit{Maruyama} showed if \(s=g\) there is a one-dimensional family of maximal subbundles (meaning bundles which achieve the bound). Maruyama then conjectured that if \(s(E)<g\) and \(E\neq L\oplus L\) then there are only finitely many maximal subbundles. \textit{Lange} and \textit{Narasimhan} found a counter example on smooth plane curves of degree 4, and further examples of the following form. \(X\) and \(Y\) are curves with a nontrivial morphism \(X@>\pi>>Y\). \(F\) is a vector bundle over \(Y\) with a one-dimensional family of maximal subbundles (e.g. \(s=g(Y)\)). Now set \(E=\pi^*F\), if \(g(X)\gg 0\) and \(\pi\) does not factor, the maximal subbundles of \(F\) pull back to maximal subbundles of \(E\), and hence \(E\) has infinitely many maximal subbundles. The paper under review shows that if \(E\) is a bundle over \(X\) with \(s(E)=s\), \(s(2s-1)<g= g(X)\) and infinitely many maximal subbundles, then there is a nontrivial morphism \(X@>\pi>>Y\) and a bundle \(F\) on \(Y\) with infinitely many maximal subbundles so that (up to twisting by a line bundle) \(E=\pi^*F\) and the maximal subbundles of \(E\) are pull backs of the maximal subbundles of \(F\). The paper then produces more examples (which are not pull backs) on smooth plane curves. Finally, the paper shows that if \(s\) is even and \(g\geq s\) there is a curve \(X\) of genus \(g\), and a vector bundle \(E\) over \(X\) with \(s(E)=s\) having infinitely many maximal subbundles.