Plane algebraic curves with many cusps, with an appendix by Eugenii Shustin

Abstract

The maximum number of cusps on a plane algebraic curve of degree d is an open classical problem that dates back to the nineteenth century. A related open problem is the asymptotic value of the maximum number of cusps on plane curves of degree d, divided by d^2, when d tends to infinity. In this paper, we improve the best known lower bound for the asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cusps are constructed too. The Appendix to this paper is devoted to the case of degree 11 and it is due to E. Shustin

Heegaard Floer Homology and Rational Cuspidal Curves

“…It says that b ď" 125`?73 432 d 2 . In [6], there is constructed a family of curves with b " 2567 8640 d 2 (for some infinite sequence of d).…”

Plane algebraic curves with prescribed singularities

“…The general genus formulas ( 6) and (7) were first proved by Hironaka [38,Theorem 2] using the resolution of the singularities of C (he defines g(C) as s i=1 g(C i )). If C ⊂ P 2 is a plane curve of degree d > 0, then C is connected (by Bezout's theorem) and we have (8) p…”

“…Now, if C ⊂ P 2 is reduced and irreducible, then C is smooth and connected and the geometric genus g(C) = g(C) is non-negative. The formulas (6) and (8) imply the genus formula…”

Injective linear series of algebraic curves on quadrics

“…For cuspidal curves, a classical open problem is to determine the maximum number of cusps realizable on a plane curve of degree d; recent asymptotic results were proven by Calabri, Paccagnan, and Stagnaro [6]. Interestingly, Koras and Palka [19] showed that complex plane rational cuspidal curves possess at most four singular points.…”