Weierstrass loci for vector bundles on curves (Q1922060)

Let \(X\) be a smooth complex projective curve, \(E\) a rank \(r\) vector bundle on \(X\) and \(V\subseteq H^0 (X, E)\) a vector space spanning \(E\). \(V\) defines a morphism \(\pi: X\to G(R, n)\), \(n:= \dim (V)\), from \(X\) to a Grassmannian. \textit{D. Perkinson} in his thesis [Trans. Am. Math. Soc. 347, No. 9, 3179-3246 (1995)], introduced the notion of higher order derived bundles and higher order torsion sheaves of \(\pi\) and studied the relations between these notions and the principal part bundles of the pair \((E, V)\). He showed that the ranks of the derived bundles (called ``differential ranks'') give a local normal form of \(\pi\) at a general point of \(X\). In our paper we use his main results to show how often there are examples of \(X\), \(E\) and \(V\) such that the higher order differential ranks are as small as possible, compatibly with the integers \(r\) and \(n\) (e.g. they vanish if \(2r\geq n\)).