# Counting curves with local tangency constraints

## TL;DR

This paper introduces new invariants for counting rational curves with local tangency constraints in symplectic manifolds, providing recursive formulas and computations related to Gromov--Witten invariants and punctured curves.

## Contribution

It develops a framework for invariants counting curves with local tangency constraints and relates them to existing Gromov--Witten invariants through recursive formulas.

## Key findings

- Derived formulas for invariants as point constraints coalesce in dimension four.
- Computed all invariants in terms of Gromov--Witten invariants of blowups.
- Studied invariants counting punctured curves with negative ends on ellipsoids.

## Abstract

We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency constraints. We give a formula describing how these invariants arise as point constraints are pushed together in dimension four, and we use this to recursively compute all of these invariants in terms of Gromov--Witten invariants of blowups. As a key tool, we study analogous invariants which count punctured curves with negative ends on a small skinny ellipsoid.

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Source: https://tomesphere.com/paper/1906.02394