Existence of Hypersurfaces with Prescribed Simple Singularities

Abstract

We derive asymptotically proper sufficient conditions for the existence of hypersurfaces W ⊂ ℙ ⁿ with prescribed simple singularities. By first constructing hypersurfaces with one simple singularity of type S corresponding to a T-smooth germ of V _d (S), d = deg(W), and then applying a version of Viro's method we show how to construct hypersurfaces of degree d with simple singularities of types S ₁,…, S _r whenever

where α > 0 is an explicit constant depending only on the dimension n.

Keywords:

• Isolated hypersurface singularities
• Prescribed singularities
• Simple singularities
• Equianalytic families
• Viro method

Acknowledgments

I would like to thank Prof. E. Shustin (Tel Aviv University), Prof. G.-M. Greuel, and Dr. C. Lossen (University of Kaiserslautern) for many fruitful discussions and suggestions.

The methods used in this paper are based to some extent on a preprint of Prof. E. Shustin, now published in Shustin (2000), and I am very grateful to him for providing his material before publication. The author is supported by grant GR-640/9-1 DFG-Schwerpunkt, “Globale Methoden in der komplexen Geometrie.”