Chern character of degeneracy loci and curves of special divisors (Q1078263)

Let X be a smooth algebraic variety, E and F vector bundles over X of ranks e and f, respectively. For a bundle map \(\phi:\quad E\to F,\) let I(k) be the k-singular locus of \(\phi\), i.e. \(I(k):=\{x\in X;\) rank\((\phi)_ x\leq k\}.\) The paper gives the fundamental class of I(k) as polynomial in the Chern classes of E and F in the case cod\(_ XI(k)=(e-k)(f-k)\) or \(I(k)=\emptyset\); cf. \textit{J. Harris} and \textit{L. Lu} for a similar result [Invent. Math. 75, 467-475 (1984; Zbl 0542.14015)]. As an application, the genus of the ''divisor singular loci'' of dimension r and degree d of a general curve of genus g is computed in the case \(g-(r+1)(g-d+r)=1\). If \(r=1\), \(d=m-2\), and \(g=2m+1\), that has been done by \textit{G. R. Kempf} [Compos. Math. 55, 157-162 (1985)].