Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities

TL;DR

This paper extends the formula for the Griffiths height of projective hypersurfaces with semihomogeneous singularities, using coverings and semistable models to handle complex singularities in algebraic geometry.

Contribution

It generalizes previous formulas to include semihomogeneous singularities by constructing suitable coverings and models, advancing the understanding of Griffiths heights in algebraic geometry.

Findings

Derived a formula for Griffiths height with semihomogeneous singularities.

Constructed finite coverings to achieve semistable models with smooth components.

Extended previous results from ordinary double points to more general singularities.

Abstract

This paper establishes the formula for the stable Griffiths height of the middle-dimensional cohomology of a pencil of projective hypersurfaces $H$, with semihomogeneous singularities, over some smooth projective curve $C$, that appears as Theorem 5.1 in the first part of this paper (arxiv:2506.15334). The proof of this formula relies on the strategy developed in my previous work (arxiv:2212.11019v3) to derive an expression for this Griffiths height when the only singularities of the fibers of $H$ over $C$ are ordinary double points. To deal with general semihomogeneous singularities, we complement this strategy by the construction of a finite covering $C'$ of $C$ such that the pencil $H' = H \times_C C'$ over $C'$ admits a smooth model $\widetilde{H}'$ with semistable fibers with smooth components. This allows us to circumvent the delicate issue of the determination of the elementary exponents attached to the singular fibers of $H/C$.