Stable maps to Looijenga pairs built from the plane

TL;DR

This paper surveys methods for counting rational curves on certain log Calabi-Yau surfaces derived from the plane, focusing on their intersection properties with divisors.

Contribution

It provides a systematic approach to counting rational curves on Looijenga pairs constructed from the projective plane.

Findings

Develops a counting technique for rational curves with maximal tangency

Applies the method to four specific log Calabi-Yau surfaces

Offers insights into the geometry of Looijenga pairs

Abstract

Choosing a normal crossings anticanonical divisor of $\mathbb{P}^2$ leads to four log Calabi-Yau surfaces, three of which are Looijenga pairs. In this survey article, I describe how to count rational curves in these that intersect each component of the divisor in one point of maximal tangency.