On the variety of plane curves of degree d with \(\delta\) nodes and \(\kappa\) cusps. (English) Zbl 0702.14019

Kang, Pyung-Lyun

Trans. Am. Math. Soc. 316, No. 1, 165-192 (1989).

Let \({\mathbb{P}}^ N\) be the projective space parametrizing plane curves of degree \(d,\) and let \(V_{d,g}\) the locally closed subset of \({\mathbb{P}}^ N\) of reduced, irreducible plane curves of degree \(d\) and geometric genus \(g.\) J. Harris [Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)] has proved that \(V_{d,g}\) is irreducible. Let \(V(d,\delta,k)\) be the subvariety of \(V_{d,g}\) of those curves which do have \(d\) nodes and \(k\) ordinary cusps as their only singularities.
The author proves that \(V(d,\delta,k)\) is irreducible for \(k\leq 3\) except possibly when \(k=3\) and \(d = 5\) or 6. The main tools used in proving this result are the propositions 2.1, 4.1 and 4.2 of the paper by Harris cited above.

Reviewer: A.Del Centina

MSC:

14H10	Families, moduli of curves (algebraic)
14H45	Special algebraic curves and curves of low genus

Keywords:

irreducibility of parametrizing variety; variety parametrizing plane curves