Plane curves of minimal degree with prescribed singularities

Details Plane curves of minimal degree with prescribed singularities Plane curves of minimal degree with prescribed singularities

1997-04-11

arxiv.org/abs/alg-geom/9704010v2

Mathematics

Algebraic Geometry

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

Scientific paper

10.1007/s002220050254

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

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