Double rational normal curves with linear syzygies

Abstract:

In this note we are looking after nilpotent projective curves without embedded points, which have rational normal curves of degree d as support, are defined (scheme-theoretically) by quadratic equations, have degree 2d and have only linear syzygies. We show that, as expected, no such curve does exist in ℙ^d, and then consider doublings in a bigger ambient space. The simplest and trivial example is that of a double line in the plane. We show that the only possibility is to take rational normal curves in ℙ^d embedded further in ℙ^{2 d} and to take a certain doubling in the sense of Ferrand (cf. [5]) in ℙ^{2 d}. These double curves have the Hilbert polynomial H(t)=2dt+1, i.e. they are in the Hilbert scheme of the rational normal curves of degree 2d. Thus, it turns out that they are natural generalizations of the double line in the plane considered as a degenerated conic.The simplest nontrivial example is the curve of degree 4 in ℙ⁴, defined by the ideal (xz−y ², xu−yv, yu−zv, u ², uv, v ²). The double rational curve allow the formulation of a Strong Castelnuovo Lemma in the sense of [7], for sets of points and double points. In the last section we mention some plethysm formulae for symmetric powers.