Moduli space of genus one curves on cubic threefold
Enhao Feng

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 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 . In particular, we show that for degree , there are exactly two irreducible main components, of which one generically parametrizes free curves birational onto their images, and the other corresponds to degree covers of lines. As a corollary, we classify components of the morphism space for a general smooth genus one curve .
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