Let $X$ be a sub-variety of $\mathbb CP^n$ and let $p_X(k)$ be its Hilbert polynomial. It is well known that some basic invariants of $X$ (such as its dimension) can be read from $p_X(k)$. I am interested to know to which extent one can read connectedness of $X$ from $p_X(k)$. Here are two precise questions. ($n$ is fixed).
Question 1. Is there a classification of polynomials, for which it is known that every subvariety in $\mathbb CP^n$ with given Hilbert polynomial is connected? If there is no such a classification, I would like at least to know a large list of such polynomials.
Question 2. Suppose that $P$ is the Hilbert polynomial of a complete intersection on $\mathbb CP^n$ of positive dimension. Is it true that any subscheme of $\mathbb CP^n$ with Hilbert polynomial $P$ is connected?