  In this paper, we study the deformations of curves in the projective 3-space
$\mathbb P^3$ (space curves), one of the most classically studied objects in
algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a
version modified by Ph. Ellia) concerning maximal families of space curves
lying on a smooth cubic surface, assuming the quadratic normality of its
general members. We also give a sufficient condition for curves lying on a
cubic surface to be obstructed in $\mathbb P^3$ in terms of lines on the
surface. For the proofs, we use the Hilbert-flag scheme of $\mathbb P^3$ as a
main tool and apply a recent result on primary obstructions to deforming curves
on a threefold developed by S. Mukai and the author.
