Classification of SNC (p,k)-planes with Hodge structures of K3 type
Abstract
A SNC (p,k)-plane is a (ℤ/p)k-Galois cover π: X → ℙ2 with a simple normal crossing branch locus. In this setting, there exist several intermediate quotients Y1, Y2,…, Yϕp,k corresponding to subgroups of (ℤ/p)k of index p. We focus on the case where each Yi is either a surface with geometric genus $0$ or $1$, with at least one with geometric genus $1$. We classify all SNC (p,k)-planes satisfying these conditions obtaining that an extensive list of cases are minimal regular surfaces of general type with positive geometric genus. Moreover, these assumptions allow us to have strong control over the weight-$2$ Hodge structure of the minimal model of the covering surface X. Therefore, we also prove that Tate and Mumford-Tate conjectures hold for such surfaces. As an independent result of our methods, we obtain a classification of SNC (p,1)-planes with geometric genus $1$, which turn out to be precisely all K3 surfaces, and a classification of SNC (p,1)-planes with geometric genus $0$, which turn out to be instead rational surfaces.
Related
- Classification of can-Var diagrams of mixed Hodge structures and monodromic mixed Hodge modules on ℂ
- Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
- Complex and Stable Complex Structures on Real Vector Bundles over Connected Sums of Quaternionic Projective Planes
- mixed Hodge structures on the strictly upper triangular Lie algebras
- Integral Hodge structures and atoms of fourfolds
BibTeX
@misc{fallucca2026classification,
title = {{Classification of SNC $(p,k)$-planes with Hodge structures of K3 type}},
author = {Federico Fallucca and Matteo Penegini and Anna Ulivi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27961},
eprint = {2610.27961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27961},
}
RIS
TY - UNPB
TI - Classification of SNC (p,k)-planes with Hodge structures of K3 type
AU - Federico Fallucca
AU - Matteo Penegini
AU - Anna Ulivi
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27961
UR - https://arxiv.org/abs/2610.27961
AB - A SNC (p,k)-plane is a (ℤ/p)^k-Galois cover π: X → ℙ^2 with a simple normal crossing branch locus. In this setting, there exist several intermediate quotients Y_1, Y_2,…, Y_ϕ_p,k corresponding to subgroups of (ℤ/p)^k of index p. We focus on the case where each Y_i is either a surface with geometric genus $0$ or $1$, with at least one with geometric genus $1$. We classify all SNC (p,k)-planes satisfying these conditions obtaining that an extensive list of cases are minimal regular surfaces of general type with positive geometric genus. Moreover, these assumptions allow us to have strong control over the weight-$2$ Hodge structure of the minimal model of the covering surface X. Therefore, we also prove that Tate and Mumford-Tate conjectures hold for such surfaces. As an independent result of our methods, we obtain a classification of SNC (p,1)-planes with geometric genus $1$, which turn out to be precisely all K3 surfaces, and a classification of SNC (p,1)-planes with geometric genus $0$, which turn out to be instead rational surfaces.
ER -