The double cover of cubic surfaces branched along their Hessian (Q2925303)

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 answer for the general surface, he deals next with the case of the Fermat surface by relying of \textit{X. Roulleau}'s results from [Mich. Math. J. 60, No. 2, 313--329 (2011; Zbl 1225.14029)].