Push-forward of degeneracy classes and ampleness

Abstract:

Let $X$ be a projective variety and $E,F$ vector bundles on $X$. Suppose $g: X \rightarrow Y$ is a surjective map onto another variety $Y$. Let $\phi : E \rightarrow F$ be any vector bundle map and $X_{k}(\phi )$ the $k$’th degeneracy locus of $\phi$. We show that the dimension of $g(X_{k}(\phi ))$ is at least equal to \[ \min \{ {\dim }Y, { \dim }X - (\text {rank }E-k)(\text {rank }F -k) \}\] under the hypothesis that $E^{*} \otimes F$ is an ample vector bundle on $X$.