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Do there exist elliptic curves on a general 3-fold hypersurface $X_7 \subset \mathbb{P}^4$ of degree $7$?

Clemens proved that for $d \ge 8$ there are no elliptic curves on the general hypersurface $X_d \subset \mathbb{P}^4$. Likewise, Ran proved the case $d = 6$.

This paper of Ferrarese and Romagnoli proves that all such $C$ must have degree a multiple of 7 and for each degree, there can only be finitely many such curves.

However, it is not clear to me if any such curves exist. I do not believe that degree $7$ elliptic curves can appear since Gruson-Lazarsfeld-Peskine shows that any such curve must be $8$-regular and then a dimension count shows that there is no such curve on a general $X_7$.

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  • $\begingroup$ The very general hypersurface of large enough degree is “hyperbolic” (so no elliptic curves can map nontrivially to it in particular). I will look later what the precise known degree constraints are. $\endgroup$
    – Ariyan Javanpeykar
    Commented Dec 26, 2023 at 14:11

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