New proofs for technical results in "Infinitesimal invariants of mixed Hodge structures'' (arXiv:2406.17118v1)

TL;DR

This paper provides new rigorous proofs for key properties of cubic forms used in establishing the generic global Torelli theorem for Fano-K3 pairs, previously verified only with computer assistance.

Contribution

It offers formal proofs for the smoothness of cubic forms and the ideal quotient condition in the context of Fano-K3 pairs, clarifying the notion of genericity.

Findings

Cubic form C is smooth for generic pairs.

The ideal quotient (J_{F,3}:Q) equals zero for generic pairs.

Provides rigorous proofs previously obtained via computer algebra.

Abstract

Cubic forms $C$ are constructed in the work of R. Aguilar, M. Green and P. Griffiths to establish the generic global Torelli theorem for Fano-K3 pairs $(X,Y)$, where $X: F=0$ is a cubic threefold in $\mathbb{P}^4$ and $Y\in|-K_X|$ is an anticanonical smooth section of $X$ defined by a quadratic form $Q$. In this article, we prove the following two results, which were previously verified with the computer aid of Macaulay2: for a generic pair $(X,Y)$, (i) the cubic form $C$ is smooth; (2) $(J_{F,3}:Q)=0$, and thereby give a precise meaning of the word ``generic" in this context.