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Journal of the London Mathematical Society 2006 74(3):607-622; doi:10.1112/S0024610706023210
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© The London Mathematical Society

Dessins D'Enfants and Hypersurfaces with Many Aj-Singularities

Oliver Labs

Mathematik und Informatik, Gebäude E2.4, Universität des Saarlandes 66123 Saarbrücken, Germany Labs{at}math.uni-sb.de, mail{at}OliverLabs.net

Received 13 May 2005. Revision received 13 February 2006.

We show the existence of surfaces of degree d in P3(C) with approximately (3j + 2)/(6j(j+1))d3 singularities of type Aj, 2 ≤ j ≤ d – 1. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants.

Our examples improve the previously known lower bounds for the maximum number Formula(d) of Aj-singularities on a surface of degree d in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in Pn, n ≥ 5.

To conclude, we work out in detail a classical idea of Segre which leads to some interesting examples, for example, to a sextic with 36 cusps.


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