Equiclassical deformation of plane algebraic curves. (English) Zbl 0924.14012
Arnold, V. I. (ed.) et al., Singularities. The Brieskorn anniversary volume. Proceedings of the conference dedicated to Egbert Brieskorn on his 60th birthday, Oberwolfach, Germany, July 1996. Basel: Birkhäuser. Prog. Math. 162, 195-204 (1998).
In this paper the author studies equiclassical families of plane algebraic curves, that means sets of irreducible curves of a given degree \(d\), geometric genus \(g\) and class \(c\) (degree of the dual curve). Equisingular families and equisingular deformations have been studied extensively. In the paper under review an ‘equiclassical’ stratification is studied by introduction of the sets \(V_{d,g,c}\) of curves with degree \(d\), geometric genus \(g\) and class \(c\). The main result is:
If \(c\geq 2g-d+2\) then a generic member of any irreducible component of \(V_{d,g,c}\) is a curve with only nodes and cusps.
That result strengthens the sufficient condition \(c\geq 2g-1\), found by S. Diaz and J. Harris [Trans. Am. Math. Soc. 309, No. 2, 433-468 (1988; Zbl 0707.14022)].
For the entire collection see [Zbl 0890.00033].
If \(c\geq 2g-d+2\) then a generic member of any irreducible component of \(V_{d,g,c}\) is a curve with only nodes and cusps.
That result strengthens the sufficient condition \(c\geq 2g-1\), found by S. Diaz and J. Harris [Trans. Am. Math. Soc. 309, No. 2, 433-468 (1988; Zbl 0707.14022)].
For the entire collection see [Zbl 0890.00033].
Reviewer: M.Morales (Saint-Martin-d’Heres)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
| 14H20 | Singularities of curves, local rings |
| 14D15 | Formal methods and deformations in algebraic geometry |
