Moduli space of genus one curves on cubic threefold

2025/06/20 by Feng, Enhao
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.16686

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 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 Mor(E,X) for a general smooth genus one curve E.

Citations

• A higher genus circle method and an application to geometric Manin's conjecture
• Non-free curves on Fano varieties
• Non-free sections of Fano fibrations
• Moduli spaces of rational curves on Fano threefolds
• Classifying sections of del Pezzo fibrations, I
• Rational curves on prime Fano threefolds of index 1
• On exceptional sets in Manin's Conjecture
• Geometric consistency of Manin's Conjecture
• Measures of irrationality of the Fano surface of a cubic threefold
• Balanced line bundles and equivariant compactifications of homogeneous spaces
• Minimal Model Program with scaling and adjunction theory
• Elliptic curves on some homogeneous spaces
• Fano surfaces with 12 or 30 elliptic curves
• A natural smooth compactification of the space of elliptic curves in projective space
• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• Geometric stability of the cotangent bundle and the universal cover of a projective manifold
• Curves of small degree on cubic threefolds
• The connectedness of the moduli space of maps to homogeneous spaces
• The enumerative geometry of rational and elliptic curves in projective space
• Stacks of Stable Maps and Gromov-Witten Invariants
• Quantum cohomology of complete intersections
• Vector Bundles Over an Elliptic Curve

Related

• Alternate compactifications of the moduli space of genus one maps
• Stable maps of genus zero in the space of stable vector bundles on a curve
• The spaces of rational curves on del Pezzo threefolds of degree one
• Moduli spaces of rational curves on Fano threefolds
• Complete moduli for fibered surfaces