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On a class of rational cuspidal plane curves. (English) Zbl 0868.14014
The authors obtain a complete list of all rational cuspidal plane curves C (i.e. having only cusps = locally irreducible singular points) with at least 3 cusps, one of which has the (maximal possible) multiplicity $$\text{deg }C-2$$. There were only very few known examples of such curves, the simplest one being Steiner’s cuspidal quartic. The main result of this paper is that up to projective equivalence, for any $$d\geq4$$ there are exactly $$[(d-1)/2]$$ such curves of degree $$d$$. The explicit rational parametrization of these curves are given, and their singularities are described in terms of multiplicity sequences. At the end, the problem of projective rigidity of rational cuspidal plane curves is discussed.

MSC:
 14H20 Singularities of curves, local rings 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
Keywords:
rational cuspidal plane curves
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References:
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