Rational and iterated maps, degeneracy loci, and the generalized Riemann-Hurwitz formula

TL;DR

This paper extends the Riemann-Hurwitz formula to rational maps with indeterminacies and singularities, applying it to iterated maps and degeneracy loci of vector bundles, revealing new geometric insights.

Contribution

It introduces a generalized Riemann-Hurwitz formula applicable to complex maps with singularities and explores its applications to iterated maps and degeneracy loci.

Findings

Generalized Riemann-Hurwitz formula for maps with indeterminacies

Application to iterated maps with branch-like singularities

Analysis of Chern classes of determinantal varieties

Abstract

We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms.