Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.36072

Some Coble surfaces admit only Bertini involutions

Federico Pieroni
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article we show that a biregular involution $i : X \to X$ defined on an unnodal Coble surface $X$ with irreducible boundary curve $\{C\} = |- 2 K_X|$ is necessarily a lift of a Bertini involution. Then we show with counterexamples that the hypothesis of irreducibility of $C$ and unnodality of $X$ are both necessary.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Federico Pieroni. 2026-10-26. Some Coble surfaces admit only Bertini involutions. https://arxiv.org/abs/2610.36072

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tautological modular forms of level two and degree two

We show how to use divisors on the projectivized Hodge bundle to construct special vector-valued modular forms and then apply invariant theory to construct all vector-valued Siegel modular forms of level two and degree two. Thus we construct all modular forms in terms of certain basic modular forms that are intimately connected to the moduli of curves of genus two.

math.AG↗
arXiv2026

Quartic Keller Maps and Tameness in Three Variables

We study invertibility and tameness for polynomial maps of affine three-space of total degree at most four. We prove that every complex Keller map in this degree range is a polynomial automorphism and, independently, that every complex polynomial automorphism of total degree at most four is tame. The invertibility argument combines homogeneous factorization of the leading part, invariant rings of conic pencils, and residue obstructions on cubic fibers. The tameness argument uses a criterion based on a constant second directional derivative, low-degree coordinate classifications, and local tameness of plane polynomial automorphisms over a polynomial parameter ring. The invertibility conclusion extends to every field of characteristic zero.

math.AG↗
arXiv2026

Low Genus Curves on Compact Shimura Surfaces

We investigate and compare the behaviors of complex curves on two major types of compact Shimura surfaces: compact variants of Picard and Hilbert modular surfaces. They are quotients of complex hyperbolic balls corresponding to moduli problems for abelian varieties with certain endomorphism types. We prove that if we fix a genus bound, then when the defining discriminant is large enough, low genus curves on the compact variant of Picard modular surfaces do not exist, while low genus curves may persist on the compact variant of Hilbert modular surfaces only if they are special curves, which parametrize those with extra endomorphisms. These results are special cases of a complex function field analogue of uniform boundedness on endomorphism types of abelian varieties. The proof requires understanding the repulsion of special subvarieties and relies on volume estimates in hyperbolic geometry.

math.AG↗