Moduli space of genus one curves on cubic threefold

TL;DR

This paper completely describes the main components of the moduli space of genus one stable maps to a smooth cubic threefold, revealing two main types of curves for degrees five and above.

Contribution

It provides a detailed classification of the irreducible components of the Kontsevich moduli space for genus one maps to cubic threefolds, including new insights into their structure.

Findings

For degree e ≥ 5, exactly two main components exist.

One component parametrizes free curves birational onto their images.

The other component corresponds to degree e covers of lines.

Abstract

Let $X$ be a smooth cubic threefold. By invoking ideas from Geometric Manin's Conjecture, we give a complete description of the main components of the Kontsevich moduli space of genus one stable maps $\overline{M}_{1,0}(X)$. In particular, we show that for degree $e\geqslant 5$, there are exactly two irreducible main components, of which one generically parametrizes free curves birational onto their images, and the other corresponds to degree $e$ covers of lines. As a corollary, we classify components of the morphism space $\text{Mor}(E,X)$ for a general smooth genus one curve $E$.