Affine curves of degree 6 and smoothings of a non-degenerate sixth order singular point. (English. Russian original) Zbl 0679.14011

Korchagin, A. B.; Shustin, E. I.

Math. USSR, Izv. 33, No. 3, 501-520 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 6, 1181-1199 (1988).

The positions of projective curves with respect to a straight line are considered in \(RP^ 2\) instead of affine curves. The case when a straight line intersects a curve of degree \( 6\) in 6 points lying on the same oval is studied in detail. Let the family of curves \(F_ t(x,y)=0\), \(t\in [0,1]\), realize some smoothings (nonsingular perturbations) of an isolate nondegenerate singular point of multiplicity six of a curve \(F_ 0(x,y)=0\). It is proved that this smoothing is the image of some nonsingular affine curve of degree \( 6\) under a special homomorphism.

Reviewer: V.F.Ignatenko

MSC:

14H20	Singularities of curves, local rings
14B07	Deformations of singularities
14N05	Projective techniques in algebraic geometry
14H45	Special algebraic curves and curves of low genus

Keywords:

singular projective curve; smoothing