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 \cite{fer}, 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, $n \geq r$ 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 $g \leq -2$ 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 $\mathbb{P}^3$ of arbitrary genus is a smooth point of the corresponding Hilbert scheme.