Doubling rational normal curves

In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in [11], we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers (2r, g, n) where r is the degree of the support, n ≥ r is the dimension of the projective space containing the double curve, and g is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple (2r, g, n) is irreducible. Furthermore, we prove that the general double curve in the families associated to (2r, r + 1, r) and (2r, 1, 2r−1) is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus g ≤ −2 form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in ℙ3 of arbitrary genus is a smooth point of the corresponding Hilbert scheme.