On the Classification of K3-Surfaces with Nine Cusps

W. Barth

Home/Archive/arXiv

arXiv · math/9805082

On the Classification of K3-Surfaces with Nine Cusps

W. Barth

View original source ↗Read source document ↗Export metadata ↓

Abstract

By a K3-surface with nine cusps I mean a compact complex surface with nine isolated double points $A_2$, but otherwise smooth, such that its minimal desingularisation is a K3-surface. In an earlier paper I showd that each such surface is a quotient of a complex torus by a cyclic group of order three. Here I try to classify these $K3$-surfaces, using the period map for complex tori. In particular I show: A $K3$-surface with nine cusps carries polarizations only of degrees 0 or 2 modulo 6. This implies in particular that there is no quartic surface in projective three-space with nine cusps. (T. Urabe pointed out to me how to deduce this from a theorem of Nikulin.) In an appendix I give explicit equations of quartic surfaces in three-space with eight cusps.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

W. Barth. 1998-05-19. On the Classification of K3-Surfaces with Nine Cusps. https://arxiv.org/abs/math/9805082

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

KEEP EXPLORING

Related papers

arXiv2026

Good minimal models with nowhere vanishing holomorphic $1$-forms

Popa and Schnell show that any holomorphic 1-form on a smooth projective variety of general type has zeros. In this article, we show that a smooth good minimal model has a holomorphic 1-form without zero if and only if it admits an analytic fiber bundle structure over a positive dimensional abelian variety.

math.AG↗

arXiv2026

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the complex $K3$ surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

math.AG↗

arXiv2026

Algebraic Zhou valuations

In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Mustaţă conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.

math.AG↗