A Riemann-Hurwitz-Plucker formula
Abstract
We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety. Our definition of ramification is relative to an algebraic family of divisors on the target variety, and our formula is obtained using the theory of refined top Chern classes. In assigning multiplicities to ramification points, we frequently have to consider excess degeneracy loci, but we are able to show nonetheless that the multiplicities are always nonnegative, and are positive under very mild hypotheses.
Cite
@article{arxiv.1811.01279,
title = {A Riemann-Hurwitz-Plucker formula},
author = {Brian Osserman and Adrian Zahariuc},
journal= {arXiv preprint arXiv:1811.01279},
year = {2019}
}
Comments
Substantial revision in v2: relaxed the hypotheses on S, added functoriality properties, proved a sharp criterion for the (strict) positivity of multiplicities of inflection points, changed "ramification" to "inflection", improved exposition throughout. 23 pages