Curves on a double surface (Q1429242)

The aim of this paper is to study deformations of space curves in the Hilbert scheme \(H_{d,g}({\mathbb P}^3)\) of curves of degree \(d\) and genus \(g\). The idea is to study curves on a scheme \(X\) which is given by the double structure on a smooth surface \(F\subset {\mathbb P}^3\) associated to the divisor \(2F\). If \(C\subset X\) is a curve, one can associate to \(C\) the divisor \(P\subset F\) which is the largest divisor contained in \(C\cap F\) and the residual scheme \(R\) to \(C\cap F\) in \(C\). Since \({\mathcal I}_{C\cap F}= {\mathcal I}_P{\mathcal I}_Z\) for some 0-dimensional scheme \(Z\subset F\), to \(C\) is associated the triple \((Z,R,P)\). Let \(H_{z,r,p}(X)\subset H_{d,g}(X)\) be the subscheme parameterizing curves whose triple \((Z,R,P)\) has Hilbert polynomials \(z,r,p\); in the article it is shown that, under some cohomological conditions, for a curve \(C_0 \in H_{d,g}(X)\) a general deformation of its associated triple \((Z,R,P)\) lifts to a deformation \(C \in H_{d,g}(X)\) of \(C_0\). This procedure can be applied to the study of \(H_{d,g}({\mathbb P}^3)\); for example it is shown that ad hoc specializations used to prove results (e.g. connectedness) on this Hilbert schemes can be obtained this way.