ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14 Paccagnan, Diego, University of Padua, Italy Maximum number of cusps on algebraic plane curves and nodes on surfaces Let C_{d} ⊂ ℙ_{ℂ}^{2} be an irreducible algebraic curve of degree d. By the morphism (x_{0}, x_{1}, x_{2}) ↦ (x_{0}^{q}, x_{1}^{q}, x_{2}^{q}), q ∈ ℕ and q ≥ 2, sequences of curves with finite many ordinary cusps are obtained. Among them is a sequence of irreducible curves, obtained from a quartic with three cusps, of degree d=36·q^{r}, q =3, r ∈ℕ, with \frac{1019}{3456} d^{2} − \frac{d}{4} − \frac{9}{8} cusps. The leading coefficient improves Hirano’s \frac{9}{32} = \frac{972}{3456} and is rather close to the limitation from above \frac{5}{16} = \frac{1080}{3456} due to K. Ivinskis. In a forthcoming joint paper with E. Stagnaro, the generalization of the method to ℙ_{ℂ}^{3} with q = 2 transforms a cubic surface into one of degree 12 with 584 nodes improving Kreiss’ and Chmutov’s result of 576 nodes; Miyaoka’s limitation is 645.