Extension of solvable coverings across horizontal divisors over the integers

Naganori Yamaguchi

Home/Archive/arXiv

arXiv · 2610.02948

Extension of solvable coverings across horizontal divisors over the integers

Naganori Yamaguchi

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the effect of deleting a horizontal divisor from an arithmetic scheme on its étale fundamental group. Assume that the scheme is regular along the divisor, that each prime component contains a normal crossings point of residue characteristic $2$, and that the generic fibers of the boundary components remain geometrically integral after finite étale base change. We prove that the kernel of the induced surjection on fundamental groups is topologically perfect. In particular, the inclusion induces an isomorphism on maximal prosolvable quotients. We apply this result to projective hyperplane complements and to the extension of representations over complete local coefficient rings.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Naganori Yamaguchi. 2026-10-02. Extension of solvable coverings across horizontal divisors over the integers. https://arxiv.org/abs/2610.02948

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

KEEP EXPLORING

Related papers

arXiv2026

The virtual fundamental class for the moduli space of surfaces of general type

We prove that the moduli stack of index-one covers of semi-log-canonical surfaces of general type is isomorphic to the KSBA moduli stack of stable general type surfaces. Using the index-one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ cover. The $\lci$ cover, as a Deligne-Mumford stack, has only locally complete intersection singularities. We then construct the moduli stack of $\lci$ covers so that it admits a proper map to the moduli stack of surfaces of general type. Next, we construct a perfect obstruction theory on this stack and a virtual fundamental class in its Chow group. We then pushforward the virtual fundamental class from the moduli stack of lci covers to the KSBA moduli space. Thus, our construction proves Donaldson's conjecture on the existence of a virtual fundamental class for KSBA moduli spaces. The smoothing of lci covering Deligne-Mumford stacks can not control all the non lci slc singularities. Thus, we construct the moduli stack of birational lci enhancement Deligne-Mumford stacks for all non lci slc singularities. The birational lci enhancement stack is in general nonseparated. We construct a separated good moduli space from this moduli stack and a perfect obstruction theory on it. A tautological invariant is defined by integrating a power of the first Chern class of the CM line bundle over the virtual fundamental class. This serves as a generalization of the tautological invariants defined by integrating tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.

math.AG↗

arXiv2026

Categorical Nielsen realization problems for generic K3 surfaces

For a complex projective K3 surface of Picard number one, we prove that every nontrivial finite subgroup of autoequivalences has order two and compute the number of conjugacy classes of such subgroups. We obtain similar formulas for the subgroups which are finite up to shifts, and deduce that such a K3 surface has an associated cubic fourfold if and only if it has an autoequivalence of order three modulo shifts. These results are proved by showing that every such subgroup fixes a Bridgeland stability condition up to the $\mathbb{C}$-action. We also establish similar existence results for curves, twisted abelian surfaces, generic twisted K3 surfaces, and standard autoequivalences of surfaces.

math.AG↗

arXiv2026

On the uniform positivity of $F$-signature under reduction modulo $p$

Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. In this paper, we give an affirmative answer to this conjecture in the case of pure subrings of regular local rings--for example, reductive quotient singularities. We also show that the conjecture can be reduced to the Gorenstein case. Finally, we discuss the connection with $F$-alpha invariants--a characteristic $p$ analog of Tian's alpha invariants introduced by Pande--for log Fano pairs.

math.AG↗