Dimension theory and degenerations of de Jonqui\`eres divisors

TL;DR

This paper confirms the validity of de Jonqui extquotesingle eres formulas for counting special divisors on general curves by using degeneration techniques, establishing their correctness in expected cases.

Contribution

It proves the de Jonqui extquotesingle eres formulas are accurate for general curves through degeneration methods, resolving a longstanding question.

Findings

De Jonqui extquotesingle eres formulas are valid for general curves.

The space of de Jonqui extquotesingle eres divisors has expected dimension in general.

Negative expected dimension implies non-existence of such divisors.

Abstract

This paper aims at settling the issue of the validity of the de Jonqui\`eres formulas. We consider the space of divisors with prescribed multiplicity, or de Jonqui\`eres divisors, contained in a linear series on a smooth projective curve. Assuming zero expected dimension of this space, the de Jonqui\`eres formulas compute the virtual number of de Jonqui\`eres divisors. Using degenerations to nodal curves we show that for a general curve equipped with a general complete linear series, the space is of expected dimension, which shows that the counts are in fact true. This implies that in the case of negative expected dimension a general linear series on a general curve does not admit de Jonqui\`eres divisors of the expected type.