Irreducibility of the Hilbert scheme of smooth curves in P-4 of degree g+2 and genus g

摘要

We denote by the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in . In this note, we show that any non-empty is irreducible, generically smooth, and has the expected dimension without any restriction on the genus g. Our result augments the irreducibility result obtained earlier by Iliev (Proc Am Math Soc 134:2823-2832, 2006), in which several low genus cases have been left untreated.