Betti Numbers of Curves and Multiple-Point Loci

TL;DR

This paper constructs cycles on Hurwitz space to relate Betti numbers of canonical curves to minimal pencils, providing a geometric interpretation and analyzing generic transversality conditions.

Contribution

It introduces a new construction of cycles on Hurwitz space and links Betti table extremal Betti numbers to minimal pencils of curves.

Findings

Extremal Betti number equals the number of minimal pencils.

Transversality hypotheses hold generically for two minimal pencils.

Constructs Eagon--Northcott cycles and compares them to multiple point loci.

Abstract

We construct Eagon--Northcott cycles on Hurwitz space and compare their classes to Kleiman's multiple point loci. Applying this construction towards the classification of Betti tables of canonical curves, we find that the value of the extremal Betti number records the number of minimal pencils. The result holds under transversality hypotheses equivalent to the virtual cycles having a geometric interpretation. We analyse the case of two minimal pencils, showing that the transversality hypotheses hold generically.