# Counting Curves on a Weierstrass Model

## TL;DR

This paper determines the number of rational curves on a Weierstrass elliptic Calabi-Yau threefold over lines in the base, confirming part of a conjecture using modularity and deformation theory.

## Contribution

It provides a new enumerative count of rational curves on Weierstrass models, linking modularity theorems and singularity deformation theory.

## Key findings

- Confirmed part of Huang, Katz, and Klemm's conjecture
- Derived explicit counts of rational curves over lines
- Connected modularity with enumerative geometry

## Abstract

Let $X\to \mathbb P^2$ be the elliptic Calabi-Yau threefold given by a general Weierstrass equation. We answer the enumerative question of how many discrete rational curves lie over lines in the base, proving part of a conjecture by Huang, Katz, and Klemm. The key inputs are a modularity theorem of Kudla and Millson for locally symmetric spaces of orthogonal type and the deformation theory of $A_n$ singularities.