Pointed Castelnuovo numbers

TL;DR

This paper derives a compact formula for counting linear series with prescribed ramification on general curves, extending classical Castelnuovo numbers and applying it to enumerative problems in moduli spaces.

Contribution

It introduces a new determinantal formula for counting linear series with ramification, generalizing Castelnuovo numbers and solving related enumerative problems.

Findings

Derived a compact determinantal formula for linear series counts

Extended classical Castelnuovo numbers to ramification conditions

Applied the formula to solve enumerative problems on moduli spaces

Abstract

The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series of dimension one, the Castelnuovo numbers specialize to the better known Catalan numbers. Using the Fulton-Pragacz determinantal formula for flag bundles and combinatorial manipulations, we obtain a compact formula for the number of linear series on a general curve having prescribed ramification at an arbitrary point, in the case when the expected number of such linear series is finite. The formula is then used to solve some enumerative problems on moduli spaces of curves.