Varieties of cuspidal curves in \({\mathbb{P}}^ r\) (Q1113958)

It is proved that the variety of rational curves of degree d in \({\mathbb{P}}^ r\) with k ordinary cusps is irreducible if the Brill-Noether number \((r+1)d-rk-r(r+1)\) is non-negative. The proof uses the case in which the Brill-Noether number is zero [\textit{D. Eisenbud} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 20, 65-87 (1987; Zbl 0625.14013)] and some other results of the same authors [Invent. Math. 74, 371-418 (1983; Zbl 0527.14022)] about the dimension and reducedness of those varieties. The other ingredient is a degeneration of rational curves to reducible ones. As a consequence, it is also proved, assuming \((r+1)d-rg-r(r+1)\) non- negative, the irreducibility of the variety of curves of degree d in \({\mathbb{P}}^ r\) of arithmetic genus g having k ordinary cusps and degenerations to general g-cuspidal rational curves.