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arXiv · alg-geom/9704010

Plane curves of minimal degree with prescribed singularities

Gert-Martin Greuel·Christoph Lossen·Eugenii Shustin
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Abstract

We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $μ(S_1)+...+μ(S_n) \leq ad^2$, there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S_1,...,S_n, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree $d \leq 14\sqrt{μ(S)}$ having a singular point of type S.

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Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin. 1998-07-08. Plane curves of minimal degree with prescribed singularities. https://doi.org/10.1007/s002220050254

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