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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2506.22126 |
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| _version_ | 1866916813018759168 |
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| author | Mordant, Thomas |
| author_facet | Mordant, Thomas |
| contents | 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 $H/C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities Mordant, Thomas Algebraic Geometry 14D07, 14L24 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 $H/C$. |
| title | Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities |
| topic | Algebraic Geometry 14D07, 14L24 |
| url | https://arxiv.org/abs/2506.22126 |