On quartic surfaces and sextic curves with singularities of type \(\tilde E_ 8\), \(T_{2,3,7}\), \(E_{12}\). (English) Zbl 0601.14031
The paper studies the normal quartic surfaces in \({\mathbb{P}}^ 3\) and the reduced sextic curves in \({\mathbb{P}}^ 2\) with a given singularity of one of the following type: a simple elliptic singularity \(\tilde E_ 8\), or a cusp singularity \(T_{2,3,7}\), or a unimodular exceptional singularity \(E_{12}\). The results obtained are of two types: (i) the description of the other singularities of the quartic surface or the sextic curve, or (ii) the existence of such surfaces or curves having prescribed singularities. The method used by the author relies heavily on Looijenga’s paper [E. Looijenga, Ann. Math., II. Ser. 114, 267-322 (1981; Zbl 0509.14035)] where a Torelli-type theorem for rational surfaces with effective anti-canonical divisors is established. Indeed, the author reduces himself to the study of double covers of \(P^ 2\) branched along a sextic curve, and proves that the surfaces in question are rational. Then he applies Looijenga’ methods to construct the moduli space for them and examines closely the action of the Weyl group on the moduli space.
Reviewer: L.Bădescu
MSC:
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14H20 | Singularities of curves, local rings |
| 14J25 | Special surfaces |
| 14B05 | Singularities in algebraic geometry |
Keywords:
Dynkin diagrams; normal quartic surfaces; sextic curves; prescribed singularities; action of the Weyl group on the moduli spaceCitations:
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