The double cover of cubic surfaces branched along their Hessian
type: Item
instance of: scholarly article
title: The double cover of cubic surfaces branched along their Hessian
zbMATH Open document ID: 1307.14013
DOI: 10.1142/S0129167X1450075X
author: Atsushi Ikeda
published in: International Journal of Mathematics
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.4242
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14J29
zbMATH DE Number: 6359568
zbMATH Keywords: cubic surfaces
zbMATH Keywords: branched coverings
zbMATH Keywords: Hodge structure
zbMATH Keywords: Néron-Severi lattices
MaRDI profile type: Publication
arXiv ID: 1012.4242
cites work: The complex hyperbolic geometry of the moduli space of cubic surfaces
cites work: The intermediate Jacobian of the cubic threefold
cites work: On the periods of certain rational integrals. I, II
cites work: On deformations of quintic surfaces
cites work: Algebraic surfaces of general type with small \(c^2_1\). III
cites work: The Varieties of Tangent Lines to Hypersurfaces in Projective Spaces
cites work: Fano surfaces with 12 or 30 elliptic curves
cites work: THE GEOMETRY OF THE FANO SURFACE OF A NONSINGULAR CUBIC $F\subset P^4$ AND TORELLI THEOREMS FOR FANO SURFACES AND CUBICS
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review text: Consider a nonsingular surface \(X\subset{\mathbb{P}^{3}}\), in general the tangent plane at a point \(P\) cuts \(X\) along a nodal curve, which is singular at \(P\) and has two proper tangent lines there. In this way one finds a double cover \(Y'\) of \(X\) and the branching \(B\) is the intersection of \(X\) with its Hessian surface.NEWLINENEWLINE When \(X\) is a cubic surface then \(B\) is singular precisely along the Eckardt points of \(X\), such a point has the property that it is contained in \(3\) lines of \(X\). The main object of the present paper is to investigate the Hodge structure of the desingularization \(Y\) of \(Y'\).NEWLINENEWLINE The procedure requires the study of the infinitesimal variations of the structure, this is done following Griffith's classical method. The author needs to construct a convenient generalized Jacobian ring, a study which was begun by him in a previous article [RIMS Kôkyûroku Bessatsu B9, 115--125 (2008; Zbl 1203.14009)]. Here, he completes the job, by combining modern homological methods with computations related to higher polars. The main outcome of this section of the paper says that \(\mathrm{NS}(Y)\) is of rank \(28\) in general, on the other hand when \(X\) contains Eckardt points then the rank is larger. In the extremal case of the Fermat surface the fact that the number of Eckardt points is largest is reflected in the result that the rank is now \(44\).NEWLINENEWLINE The author also determines the integral Hodge structure and the Néron-Severi lattices on \(Y\) by investigating the relations between the geometry of \(Y\) and the properties of the Fano surface \(F\) of lines on the cubic threefold \(V\) which is the triple cover of \( {\mathbb{P}^{3}}\) branched along \(X\). His careful work depends on results about the Galois action on the cohomology of \(V\), hence of \(F\), which are due to \textit{D. Allcock} et al. [J. Algebr. Geom. 11, No. 4, 659--724 (2002; Zbl 1080.14532)]. In this way, the author gives an answ…
reviewed by: Alberto Collino
description: scientific article; zbMATH DE number 6359568