Vol. 109, No. 2, 1983

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On the singularities of almost-simple plane curves

Tibor Bisztriczky

Vol. 109 (1983), No. 2, 257–273

Let Γ be a differentiable curve in a real projective plane P2 intersected by every line at a finite number of points. A point of Γ is ordinary if Γ is locally convex at that point; otherwise, the point is singular. Let the singular points of Γ consist of n1 inflections, n2 cusps (cusps of the first kind) and n3 beaks (cusps of the second kind). Then Γ is singular if n(Γ) = n1 + n2 + n3 > 0; otherwise, Γ is non-singular. The following questions arise naturally: Under what conditions is Γ singular? What is then the minimum value of n(Γ) and is it dependent or independent of the type of singularities that Γ may possess? Presently, we determine a class of curves Γ for which n(Γ) 2 but n1 + 2n2 + n3 4.

Mathematical Subject Classification 2000
Primary: 53C70
Secondary: 53A20, 53C75
Received: 28 May 1981
Revised: 10 February 1982
Published: 1 December 1983
Tibor Bisztriczky