Some Singular Curves and Surfaces Arising from Invariants of Complex Reflection Groups

Abstract

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups. As a result, we improve some lower bounds on the number of singularities of a given type that a plane curve or a surface in ${\mathbf{P}}^3(\mathbb{C})$ of a given degree might have.

Keywords:

• Singularities
• reflection groups
• curve
• surface

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Acknowledgments

I wish to thank warmly Alessandra Sarti, Oliver Labs and Duco van Straten for useful comments and references and Gunter Malle for a careful reading of a first version of this paper. Figures were realized using the software SURFER [imaginary].

Notes

1 There is an important exception to this remark: all the singular points of the surface of degree 8 with 48 singularities of type D₄ constructed in Example 5.3 have rational coordinates.

2 Some Milnor and Tjurina numbers were computed with SINGULAR [Decker et al. 18].

Additional information

Funding

This article is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. The hidden computations which led to the discovery of the polynomial g were done using the High Performance Computing facilities of the MSRI. The author is partly supported by the ANR (Project No ANR-16-CE40-0010-01 GeRepMod)