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Geometric Manin's Conjecture for Fano 3-Folds

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arxiv 2209.05517 v2 pith:7YXHTWXJ submitted 2022-09-12 math.AG

Andrew Burke , Eric Jovinelly This is my paper · ORCID
classification math.AG
keywords curvesfanofreerationalclassconjecturegeometricirreducible
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We classify families of free rational curves on all smooth Fano threefolds over the complex numbers. In particular, we prove the family of very free rational curves representing any fixed numerical curve class is either irreducible or empty. This proves Geometric Manin's Conjecture in dimension three. For general Fano threefolds of each deformation type, our results allow us to explicitly count the number of components of the moduli space of irreducible, geometrically rational curves, which may not be free, representing any numerical class.

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  1. Homological stability and weak approximation

    math.AG 2025-09 conditional novelty 7.0 of 10

    Homological stability holds for jet-constrained spaces of sections of projective bundles, conic bundles, and quadric surface bundles over a curve.

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