Fix $k>1$ and $d>2$. Is there any known estimate on the minimal number $n=n(k,d)$ such that in $\mathbb CP^n$ any smooth submanifold of codimension $k$ and degree $d$ is a complete intersection? I am curious in particular what is known for small $d$ (i.e. $d=3,4,5...$).
This question is of course related to Hartshorne's conjecture, but I guess it should be much easier.
ADDED. Mahdi's comment settles the co-dimension two case ($k=2$). I wonder for $k>2$ is it known at least that $n(k,d)$ is finite for all $d$?