Geodesic
On the asymptotic enumerativity property for Fano manifolds
Forum of Mathematics, Sigma, Tome 12 (2024)

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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 $d \leq (n+3)/3$ in ${\mathbb P}^n$. 
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     author = {Roya Beheshti and Brian Lehmann and Carl Lian and Eric Riedl and Jason Starr and Sho Tanimoto},
     title = {On the asymptotic enumerativity property for {Fano} manifolds},
     journal = {Forum of Mathematics, Sigma},
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Roya Beheshti; Brian Lehmann; Carl Lian; Eric Riedl; Jason Starr; Sho Tanimoto. On the asymptotic enumerativity property for Fano manifolds. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.87

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