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Cuspidal quintics and surfaces with $p_{g}=0$, $K^{2}=3$ and 5-torsion

Published online by Cambridge University Press:  01 February 2016

Carlos Rito*
Affiliation:
Universidade de Trás-os-Montes e Alto Douro (UTAD) , Quinta de Prados , 5000-801 Vila Real , Portugal email [email protected] Current address:Departamento de Matemática , Faculdade de Ciências da Universidade do Porto , Rua do Campo Alegre 687 , 4169-007 Porto , Portugal email [email protected]

Abstract

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If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.

We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

Type
Research Article
Copyright
© The Author 2016 

References

Barth, W., ‘A quintic surface with 15 three-divisible cusps’, Preprint, Erlangen, 2000.Google Scholar
Barth, W. and Rams, S., ‘Equations of low-degree projective surfaces with three-divisible sets of cusps’, Math. Z. 249 (2005) no. 2, 283295.CrossRefGoogle Scholar
Barth, W. and Rams, S., ‘Cusps and codes’, Math. Nachr. 280 (2007) no. 1–2, 5059.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235265.Google Scholar
Catanese, F., ‘Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications’, Invent. Math. 63 (1981) 433465.Google Scholar
van der Geer, G. and Zagier, D., ‘The Hilbert modular group for the field ℚ(√13)’, Invent. Math. 42 (1977) 93133.Google Scholar
Tan, S.-L., ‘Surfaces whose canonical maps are of odd degrees’, Math. Ann. 292 (1992) no. 1, 1329.Google Scholar
Tan, S.-L., ‘Cusps on some algebraic surfaces and plane curves’, Complex Analysis, Complex Geometry and Related Topics – Namba 60 (2003) 106121.Google Scholar