On the Irreducibility of the Hilbert Scheme of Curves in ℙ 5

Abstract

Denote by ℐ d,g,r the Hilbert scheme parametrizing smooth irreducible complex curves of degree d and genus g embedded in ℙ r . Severi (1921) claimed that ℐ d,g,r is irreducible if d ≥ g + r. Ein proved that the conjecture is true for r = 3 and 4, and in general that ℐ d,g,r is irreducible if (Ein, 1986, 1987). As it is known, for r ≥ 6 the conjecture is incorrect and r = 5 remains the only unsettled case. Here I prove that ℐ d,g,5 is irreducible, if , which doesn't yet resolve Severi's conjecture for r = 5, but expands the known irreducibility range.