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Plane curves of minimal degree with prescribed singularities. (English) Zbl 0924.14013

The authors consider the classical problem: Given an integer \(d\geq 3\) and types \(S_1,S_2, \dots,S_n\) of plane curve singularities, does there exist a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(S_1,S_2, \dots,S_n\)? The complete answer is known only for nodal curves, by Severi: An irreducible curve of degree \(d\) with \(n\) nodes as its only singularities, exists if and only if \(0\leq n\leq(d-1) (d-2)/2\). – Even for ordinary cusps, there is no complete answer. The main theorem proved in this paper is:
For any integer \(d\geq 1\) and topological types \(S_1,S_2, \dots, S_n\) of plane curve singularities, satisfying \(\sum^n_{i=1} \mu(S_i)\leq d^2/392\) there exists a reduced irreducible plane projective curve of degree \(d\) with exactly \(n\) singular points of types \(S_1,S_2, \dots,S_n\).
As a consequence the authors obtain:
For any topological type \(S\) of a single plane curve singularity there exists a reduced irreducible plane projective curve of degree \(\leq 14\sqrt {\mu(S)}\) with singularity \(S\) at the origin.

MSC:

14H20 Singularities of curves, local rings
14B05 Singularities in algebraic geometry
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