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La rationalité des schémas de Hilbert de courbes gauches rationnelles suivant Katsylo. (The rationality of the Hilbert schemes of rational skew curves according to Katsylo). (French) Zbl 0696.14002
Algebraic curves and projective geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1389, 87-90 (1989).
[For the entire collection see Zbl 0667.00008.]
Let H be the Hilbert scheme of rational curves of degree d in $${\mathbb{P}}^ n\quad (n\geq 2).$$ The paper shows that H is rational. This follows from Katsylo’s theorem on the rationality of the scheme of invariants of binary forms [P. I. Katsylo, Math. USSR, Izw. 25, 45-50 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.4, 705-710 (1984; Zbl 0593.14017)]. The proof depends on the parity of d. For $$d\quad even$$ it is invariant theoretic and for $$d\quad odd$$ it depends on the local triviality of certain conic bundles.
Reviewer: G.Horrocks
##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14H10 Families, moduli of curves (algebraic) 14M20 Rational and unirational varieties
##### Keywords:
Hilbert scheme of rational curves; rationality