Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow \mathbb{P}^n_B$, the composition of this closed immersion with the natural projection $\mathbb{P}^n_B \to B$ is $\pi$ and there exists a Hilbert polynomial $P$ such that for all $b \in B$, the fiber $\pi^{-1}(b)$ is of Hilbert polynomial $P$ in $\mathbb{P}^n_b$. Assume that $B$ is integral. Suppose that there exists a closed point $b_0 \in B$ such that the fiber $\pi^{-1}(b_0)$ can be embedded into $\mathbb{P}^3$ i.e., the closed immersion of $\pi^{-1}(b_0) \hookrightarrow \mathbb{P}^n$ factors through $\mathbb{P}^3$. Is it then possible to find an open neighbourhood $U$ of $b_0$ such that for all $u \in U$, $\pi^{-1}(u)$ can be embedded into $\mathbb{P}^3_u$ i.e., the closed immersion of $\pi^{-1}(u) \hookrightarrow \mathbb{P}^n_u$ factors through $\mathbb{P}^3_u$? If not, is there any known condition on $B$ under which this happens?

EDIT By "closed immersion $i:A \hookrightarrow \mathbb{P}^n$ factors through $\mathbb{P}^3$", I mean that there is a closed immersion $i_1:A \hookrightarrow \mathbb{P}^3$ and a LINEAR embedding $i_2:\mathbb{P}^3 \hookrightarrow \mathbb{P}^n$ such that $i=i_2 \circ i_1$.


The answer to the first question is no. In the moduli space $\mathcal{M}_{10}$ of curves of genus 10, the complete intersections $(3,3)$ in $\Bbb{P}^3$ form a strict subvariety $\mathcal{CI}$. Pick for $B$ a subvariety of $\mathcal{M}_{10}$ which intersect $\mathcal{CI}$ transversally at one point $[C]$. Embed the corresponding family $\mathcal{C}\rightarrow B$ into $\mathbb{P}^{9}_B$ by the canonical embedding (replacing $B$ by an open subset). At $[C]$ this embedding factors through $C\hookrightarrow \mathbb{P}^3$, but not at the other points of $B$.

As for the second question, no condition on $B$ will ensure what you ask. You may write down conditions on the family $\mathcal{C}\rightarrow B$, but they will be essentially tautological (existence of a line bundle on $\mathcal{C}$ giving the required embedding on each fiber.)

EDIT : The answer to the edited question is still no. Take $B=\mathbb{C}$, and consider the embedding $\mathbb{P}^1\times B\hookrightarrow \mathbb{P}^4\times B$ given by $([X:Y],t)\mapsto ([X^4:X^3Y: tX^2Y^2: XY^3:Y^4],t)$. The embedding of $\mathbb{P}^1\times \{0\} $ in $\mathbb{P}^4$ factors through $\mathbb{P}^3$, but not those of $\mathbb{P}^1\times \{t\} $ for $t\neq 0$.

  • $\begingroup$ My impression was that the question asked whether the embedding of a special curve can factor through a LINEARLY embedded ${\mathbb P}^3$, and in this example it is embedded via the second Veronese embedding. $\endgroup$ – Sasha Apr 16 '15 at 14:48
  • $\begingroup$ I don't see anything linear in "the closed immersion of $\pi ^{-1}(b_0)\hookrightarrow \mathbb{P}^n$ factors through $\mathbb{P}^3$". Of course this would be a completely different question. $\endgroup$ – abx Apr 16 '15 at 15:38
  • $\begingroup$ @abx: Thank you very much for your answer. But, I am sorry, I meant linearly embedded $\mathbb{P}^3$. I will edit the original question. $\endgroup$ – Ron Apr 16 '15 at 16:00

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.