On the asymptotic enumerativity property for Fano manifolds

Abstract

We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree $d\le (n+3)/3$ in ${\mathbb{P}}^n$.

arXiv commentminor changes, 32 pages, to appear in Forum of Mathematics, Sigma

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• The intrinsic normal cone
• Gromov-Witten invariants in algebraic geometry
• The moduli space of stable quotients
• Geometric Manin's Conjecture and rational curves
• Spaces of rational curves in complete intersections
• The quantum cohomology of blow-ups of P^2 and enumerative geometry
• Interpolation for Restricted Tangent Bundles of General Curves
• Virtual classes of Artin stacks
• Tevelev degrees in Gromov-Witten theory
• Fixed-domain curve counts for blow-ups of projective space