arXiv subjects

Thomas Mordant

Publications and source records attributed to Thomas Mordant.

4 recordsLinked to original sources

arXiv2025

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

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 $H/C$.

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arXiv2025

Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. I

We study the Griffiths heights associated to the middle-dimensional cohomology of pencils of projective hypersurfaces, by comparing them to heights defined by means of geometric invariant theory (GIT). Kato and Koshikawa have conjectured a Northcott property for the Kato heights attached to motives over number fields, and investigated its consequences. Bounding these Griffiths heights in terms of the GIT heights would constitute a geometric counterpart, valid over function fields of characteristic zero, of Kato and Koshikawa's conjecture. Part of our results follows from our earlier works on the computation of these Griffiths heights in the case of pencils with generic singularities, and on semistability criteria for singular projective hypersurfaces, combined with a general formalism of GIT heights over function fields. We also establish estimates between the Griffiths and GIT heights associated to pencils of projective hypersurfaces, which are valid beyond the case of generic singularities. To achieve this, we establish diverse results of independent interest. Notably we extend our previous computations of Griffiths heights to pencils of projective hypersurfaces with semihomogeneous singularities, and we show the lower semicontinuity of the stable Griffiths height attached to polarized variations of Hodge structures over complex algebraic curves.

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arXiv2024

Griffiths heights and pencils of hypersurfaces

The Griffiths height of a variation of Hodge structures over a projective curve is defined as the degree of its canonical line bundle, as defined by Griffiths and generalized by Peters to allow bad reduction points. It may be seen as a geometric analog of the Kato height attached to pure motives over number fields. In this paper, we establish various formulas expressing the Griffiths height of the middle-dimensional cohomology of a pencil of projective complex hypersurfaces in terms of characteristic classes.

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arXiv2024

A note on the semistability of singular projective hypersurfaces

In this note, we give sufficient conditions for the (semi)stability of a hypersurface $H$ of $\mathbb{P}^N_k$ in terms of its degree $d$, the maximal multiplicity $δ$ of its singularities, and the dimension $s$ of its singular locus. For instance, we show that $H$ is semistable when $d \geq δ\min (N+1, s+3)$. The proof relies in particular on Benoist's lower bound for the dimension of the intersection of the singular locus $H_{\mathrm{sing}}$ of $H$ with some linear subspace of $\mathbb{P}^N_k$ associated to a one-parameter subgroup $λ$ of $\mathrm{SL}_{N+1, k}$, in terms of the numerical data in the Hilbert-Mumford criterion applied to $λ$ and to an equation $F_H$ of $H$.

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