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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?

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Let me give a negative answer to Question 2, which shows that the answer to Question 1 should be almost always negative (perhaps with very few exceptions). The Hilbert polynomial of a complete intersection of type $(2,2)$ in $\mathbb{P}^3$ is $4k$. But this is also the Hilbert polynomial of the union of a plane quartic curve (embedded in $\mathbb{P}^3$) and two points lying anywhere in $\mathbb{P}^3$.

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    $\begingroup$ Thank you for this answer. It would be really curios then to see an approximate list of exceptions. $\endgroup$ – aglearner Feb 7 '14 at 13:30
  • $\begingroup$ For curves in $\Bbb{P}^n$ I believe the only exceptions are indeed plane curves. $\endgroup$ – abx Feb 7 '14 at 17:42

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