Question

Let $Z$ be a local complete intersection subscheme of dimension $m$ in $\mathbb{P}^{2m+1}$. Let $P$ be the Hilbert polynomial of $Z$. Denote by $Hilb_P$ the Hilbert scheme of local complete intersection subschemes with Hilbert polynomial $P$. Is it true that for a generic subscheme $Z'$ in $Hilb_P$, $I(Z')$ can be generated by polynomials that define smooth hypersurfaces in $\mathbb{P}^{2m+1}$ i.e., does there exists polynomials $P_1, ..., P_n$ such that $I(Z')=(P_1,...,P_n)$ and the zero locus of $P_i$ is smooth for all $i$?

A slightly weaker condition would be to ask if a generic hypersurface in $pr_2 Hilb_{P,Q}$ is smooth where $Hilb_{P,Q}$ is the flag Hilbert scheme of pairs $(Z \subset X)$ where $Z$ is a local complete intersection subscheme of dimension $m$ with Hilbert polynomial $P$ contained is a hypersurface $X$ in $\mathbb{P}^{2m+1}$ with Hilbert polynomial $Q$ and $pr_2$ denotes the natural projection map.

Answer 1

I believe that the answer to the first question is no for the reason that a local complete intersection is pretty far from a complete intersection. Take $Z$ to be $3$ points in $\mathbb P^3$ which aren't on a line. Then the Hilbert scheme of local complete intersections is a dense open subset of the Hilbert scheme of $3$ points in $\mathbb P^3$. However, the complete intersections will have to be the intersection of a cubic hypersurface with two hyperplanes, for degree reasons. Thus, the complete intersections will all lie on a single line in $\mathbb P^3$, which is not true for 3 generic points.

If you assumed that $Z$ is a complete intersection, not just a local complete intersection, then what you want is probably true.