Enumeration of Elliptic Curves via Elliptic Gromov-Witten Invariants of Four Dimensional Projective Fano Hypersurfaces
Masao Jinzenji, Ken Kuwata
TL;DR
This work investigates how elliptic Gromov–Witten invariants of four-dimensional projective Fano hypersurfaces relate to counts of elliptic curves. The authors identify degenerate contributions in genus $1$ invariants, express them via genus $0$ data, and propose a conjectural formula that adds an integer count $E_{d,a,b,c}$ of elliptic curves to complete the invariant count, enabling a closed, automatable computation. They compute genus $0$ and genus $1$ invariants for the fourfolds $M_6^k$ with $k=1, 2, 3, 4, 5$ up to degree $d\le 5$, find $E_{d,a,b,c}$ nonnegative, with $E_{1,\cdot}=E_{2,\cdot}=0$ and $E_{3,\cdot}$ matching Katz's counts; for degree $3$ curves they provide a moduli-space construction that yields $\tilde{E}_{3,a,b,c}$ which agrees with $E_{3,a,b,c}$. The paper thus offers a practical framework for elliptic-curve enumeration in fourfolds, connecting GW theory, BCOV-type structures, and classical enumerative geometry through explicit integral expressions on moduli spaces. The results point toward a scalable approach to counting elliptic curves in Fano fourfolds via elliptic GW invariants and associated degeneracy corrections.
Abstract
In this note, we propose a conjecture that clarifies the relationship between the number of degree d elliptic curves in complex four-dimensional projective Fano hypersurfaces and their degree d elliptic Gromov-Witten (GW) invariants. The elliptic GW invariants are computed using the elliptic virtual structure constants proposed in our previous works.