On enumerative problems for maps and quasimaps: freckles and scars

TL;DR

This paper develops a new approach to counting maps between projective spaces by embedding them into a space of quasimaps, introducing concepts like freckles and scars, and proposing a conjecture for computing these counts via integrals.

Contribution

It introduces a novel method using quasimaps to count maps, incorporating freckles and scars, and formulates the smooth conjecture for calculating these enumerative invariants.

Findings

Freckle/scar calculus effectively computes intersection counts.

Embedding maps into quasimaps simplifies enumerative problems.

The smooth conjecture offers a potential integral formula for counting maps.

Abstract

We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own. When a quasimap is not a map, it contains freckles (studied earlier) and/or scars, appearing when the complex dimension of the source is greater than one. We consider a lot of examples showing that freckle/scar calculus (using excess intersection theory) works. We also propose the "smooth conjecture" that may lead to computation of the number of maps by an integral over the space of quasimaps.