Plane curves of minimal degree with prescribed singularities

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 $\mu (S_1)+...+\mu (S_n)\le a{d}^2$, 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 $d\le 14\sqrt{\mu (S)}$ having a singular point of type S.

33 pages, LaTeX 2e, corrected some typos, simplified proofs of Lemmas 3.1, 4.1

Cited by in corpus (1)

• Maximal rank for planar singularities of multiplicity two