Symmetric differentials on weighted hypersurfaces
Abstract
We study symmetric differentials on weighted hypersurfaces. Let k be an algebraically closed field and let Xd⊂ ℙk(a0,…,aN), N≥3, be a weighted hypersurface, with amin:=mini ai. In characteristic zero we prove twisted vanishing in the range t<m amin for quasi-smooth hypersurfaces of degree d≥2amin, with the corresponding reflexive statement on the coarse space. We also prove a tame positive-characteristic analogue; in particular, reflexive symmetric differentials of every positive order vanish. In the second part we specialize to k=ℂ and to surfaces in weighted projective three space. Using the quotient-singularity criterion of Asega--De Oliveira--Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.
Related
- Locally symmetric hypersurfaces in globally symmetric spaces
- Remarks on the monodromy conjecture for symmetric determinantal hypersurfaces
- Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
- Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields
- Symmetric power L-functions of a weighted hyper-Kloosterman family
BibTeX
@misc{xu2026symmetric,
title = {{Symmetric differentials on weighted hypersurfaces}},
author = {Wan-Yuan Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27625},
eprint = {2609.27625},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27625},
}
RIS
TY - UNPB
TI - Symmetric differentials on weighted hypersurfaces
AU - Wan-Yuan Xu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.27625
UR - https://arxiv.org/abs/2609.27625
AB - We study symmetric differentials on weighted hypersurfaces. Let k be an algebraically closed field and let X_d⊂ ℙ_k(a_0,…,a_N), N≥3, be a weighted hypersurface, with a_min:=min_i a_i. In characteristic zero we prove twisted vanishing in the range t<m a_min for quasi-smooth hypersurfaces of degree d≥2a_min, with the corresponding reflexive statement on the coarse space. We also prove a tame positive-characteristic analogue; in particular, reflexive symmetric differentials of every positive order vanish. In the second part we specialize to k=ℂ and to surfaces in weighted projective three space. Using the quotient-singularity criterion of Asega--De Oliveira--Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.
ER -