I was reading A. de Jong and J. Starr's paper "Low degree complete intersections are rationally simply connected", which can be found at http://www.math.stonybrook.edu/~jstarr/papers/nk1006g.pdf, and I came across with the following claim in the proof of Lemma 5.5, page 23: for a general $m$-uple of points on a smooth, projective variety $X$, every irreducible curve $C$ containing these points satisfy $h^{1}(C, N_{C/X}(-m)) = 0$. The authors do not say about this fact, they just use it. I would like to know how to prove it, or a reference where I can find the proof. I am thinking it is necessary $C$ to be smooth. May anyone help me with this? I thank a lot.

twomore words. The induction argument above applies to arithmetic genus, not just geometric genus. An irreducible, projective (reduced) curve of arithmetic genus $0$ is automatically smooth of geometric genus $0$. So smoothness is automatic for the curves $C$ in that proof. At any rate, the OP is correct that de Jong and I should have spelled these things out for the reader. $\endgroup$