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Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for the universal family $\pi:\mathcal{C} \to B$ over $B$, every fiber is contained in $X$. Suppose $X$ is of degree $d$. The questions are then as follows:

1) Does there exist an open subset $U$ of $B$ such that there exists another smooth surface $X'$ in $\mathbb{P}^3$ (different from $X$) of degree $d$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$? If this is not the case can we impose any criterion (for example any generacity condition) under which this would hold true?

2) Suppose $C$ is a curve that intersects every fiber of $\pi$ transversally. Also assume that for every $b \in B$, there exists a smooth degree $d+1$ surface in $\mathbb{P}^3$ containing the curve $\pi^{-1}(b)$ and $C$. Does there exist an open subset $U$ of $B$ such that there exists a smooth surface $Y$ in $\mathbb{P}^3$ of degree $d+1$ containing the fibers $\pi^{-1}(t)$ for all $t \in U$ and $C$?

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  • $\begingroup$ Probably it would help to get answers if you give some motivation for the question (what are you after? which kind of conditions would be ok in your setting?) Something that confuses me: are you assuming Hilb_{P,Q} irreducible? If so, what does this imply on the pair, (P,Q)? Otherwise, what does "a general element" mean? $\endgroup$
    – quim
    Nov 20, 2013 at 13:42
  • $\begingroup$ @quim: The motivations is as follows: Given a linear system in a surface does there exists "small" deformations of the surface which contains an open subset of the linear system. As for the other questions: We can assume $Hilb_{P,Q}$ irreducible or take an irreducible component in here. $\endgroup$
    – Naga Venkata
    Nov 20, 2013 at 13:53
  • $\begingroup$ Do you mean, for each divisor in U there is a deformation of S which contains it? $\endgroup$
    – quim
    Nov 20, 2013 at 14:59
  • $\begingroup$ @quim: yes. I am asking whether there exist at least one non-trivial deformation of $S$ which contains all the elements of $U$. We however can take $U$ small enough if it helps. $\endgroup$
    – Naga Venkata
    Nov 20, 2013 at 15:56
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    $\begingroup$ Your (1) seems like nonsense to me. First of all, open neighborhood of $B$ in what? Second of all, curves corresponding to points of $U$ will either sweep out $X$ (if $U$ only parameterizes curves in $X$) or a dense subset of projective space. $\endgroup$
    – Jack Huizenga
    Nov 23, 2013 at 5:46

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