New proofs for technical results in "Infinitesimal invariants of mixed Hodge structures'' (arXiv:2406.17118v1)
Zhenjian Wang
TL;DR
The paper provides a purely algebraic refinement of the genericity notion for the cubic-form construction underlying the generic global Torelli theorem for Fano–K3 pairs. By analyzing Milnor algebras, defects of linear systems, and the strong Lefschetz property, it replaces computer-assisted verifications with explicit open-set arguments: (i) the open set $\mathcal{U}\subset S^3V$ where the associated cubic form $C$ is smooth is nonempty, and (ii) for a general cubic $F$, there exists a quadratic form $Q$ such that $Y=\{F=Q=0\}$ is smooth and $(J_{F,3}:Q)=0$. The work clarifies the meaning of genericity in this geometric–algebraic setting and extends the reach of the original results beyond computational checks, leveraging Lefschetz theory to guarantee the desired injectivity for generic $Q$. This strengthens the theoretical foundation for infinitesimal invariants of mixed Hodge structures in this context.
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.