Ramification of the moduli map for hyperplane sections of K3s and hypersurfaces
Abstract
Finiteness and injectivity of the moduli morphisms μn:|nH|sm→ℳgn associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the multiple is sufficiently positive, so our focus is on low-degree phenomena. In particular, for a general K3 surface of degree $2$, we prove that the primitive moduli map is quasi-finite but ramified at exactly $171$ points, identified with the jumping lines of a logarithmic tangent bundle associated with the branch sextic. On the other hand, we prove that if X⊂ℙn is a general hypersurface of degree at least $4$, then H0(Y,TX|Y)=0 for every smooth hyperplane section Y of X. In particular, the moduli map for smooth hyperplane sections of a general quartic K3 surface is unramified. We moreover show that ramification does occur for higher multiples of the primitive polarisation on both degree-$2$ and degree-$4$ K3 surfaces, and relate the question to the stability of the tangent bundle.
Related
- Monodromy and big quantum cohomology of hyperplane sections of (co)adjoint varieties
- Lefschetz Properties of complete intersesctions and maximal variation of hyperplane sections
- The moduli space of maps from hypersurfaces and its Goodwillie tower
- Moduli of Conics on General Plucker Linear Sections of Grassmannians
- A NOTE ON THE RAMIFICATION CHARACTER, RAMIFICATION POLYNOMIALS AND ASSOCIATED DIRICHLET L-FUNCTION
BibTeX
@misc{faro2026ramification,
title = {{Ramification of the moduli map for hyperplane sections of K3s and hypersurfaces}},
author = {Dario Faro and Frank Gounelas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29501},
eprint = {2609.29501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29501},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ramification of the moduli map for hyperplane sections of K3s and hypersurfaces
AU - Dario Faro
AU - Frank Gounelas
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29501
UR - https://arxiv.org/abs/2609.29501
AB - Finiteness and injectivity of the moduli morphisms μ_n:|nH|_sm→ℳ_g_n associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the multiple is sufficiently positive, so our focus is on low-degree phenomena. In particular, for a general K3 surface of degree $2$, we prove that the primitive moduli map is quasi-finite but ramified at exactly $171$ points, identified with the jumping lines of a logarithmic tangent bundle associated with the branch sextic. On the other hand, we prove that if X⊂ℙ^n is a general hypersurface of degree at least $4$, then H^0(Y,T_X|_Y)=0 for every smooth hyperplane section Y of X. In particular, the moduli map for smooth hyperplane sections of a general quartic K3 surface is unramified. We moreover show that ramification does occur for higher multiples of the primitive polarisation on both degree-$2$ and degree-$4$ K3 surfaces, and relate the question to the stability of the tangent bundle.
ER -