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

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.

💡 Research Summary

The paper by Zhenjian Wang provides rigorous, computer‑free proofs of two technical statements that were originally verified by Macaulay2 in the work of Aguilar, Green, and Griffiths on infinitesimal invariants of mixed Hodge structures for Fano‑K3 pairs. A Fano‑K3 pair consists of a smooth cubic threefold (X={F=0}\subset\mathbb{P}^4) and a smooth anticanonical K3 surface (Y={F=Q=0}) cut out by a quadratic form (Q). The original paper needed (i) the smoothness of a certain cubic form (C) attached to the pair, and (ii) the vanishing of the colon ideal ((J_{F,3}:Q)=0). Both statements were proved only by explicit computer calculations, leaving the meaning of “generic” vague.

Wang first introduces a precise notion of genericity. Let (V) be a 5‑dimensional complex vector space, (S=\operatorname{Sym}^\bullet V) its homogeneous coordinate ring, and (J_F) the Jacobian ideal generated by the first partial derivatives of a cubic form (F\in S_3). Define \