Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

TL;DR

This paper introduces a new moduli space of complete quasimaps to a blow-up of projective space, conjectures its enumerative significance, and proves the conjecture in dimension two using Brill-Noether theory.

Contribution

It constructs a novel moduli space of complete quasimaps to blow-ups of projective space and establishes its enumerative properties in dimension two.

Findings

Spaces are pure of expected dimension.

Conjecture relates intersection numbers to curve counts with incidence conditions.

Proven in dimension two using Brill-Noether theorem for toric surfaces.

Abstract

We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.