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We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\mathbb P^3$ is a $\mathbb P^5$ bundle on $\mathbb P^3$. Can we determine the structure of the closed subscheme of $\mathrm{Hilb}^{2t+1}(\mathbb P^3)$ representing the conics contained in $X$? If it is very difficult to give a complete answer, some partial results are also welcomed, for example, if $\delta=2,3,4$.

The most interesting case for me is $\delta=3$ and being singular, and one following comment gives a complete answer to the non-singular case by the lemma of 27 lines.

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    $\begingroup$ The Hilbert scheme of conics in $\mathbb{P}^3$ is not $\mathbb{P}^5 \times \mathbb{P}^3$; in fact it is a $\mathbb{P}^5$-bundle (nontrivial!) over $\mathbb{P}^3$. $\endgroup$
    – Sasha
    Commented Oct 8, 2020 at 19:06
  • $\begingroup$ You are right. I made a stupid mistake. $\endgroup$
    – var
    Commented Oct 8, 2020 at 19:11
  • $\begingroup$ This is an easy exercise for $\delta =2$, and a not so easy one for $\delta =3$. As soon as $\delta \geq 4$, the Hilbert scheme depends heavily on the particular surface you are considering: it is empty for a general surface of degree $\delta $. $\endgroup$
    – abx
    Commented Oct 8, 2020 at 19:19
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    $\begingroup$ The most interesting case for me is $\delta=3$. Can you say anything about it? The dimension (it seems to be $1$ in general), the degree (projected into $\mathbb P^3$ or some other setting), or some other things? $\endgroup$
    – var
    Commented Oct 8, 2020 at 19:25
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    $\begingroup$ In the case $\delta=3$ - in my opinion - the most important observation is that every conic has a residual line on the hypersurface. In the other direction, given a line on the hypersurface, any plane through the line determines a residual conic. Then the projection of the Hilbert scheme of conics into $\mathbb{P}^3$ is the union of a line in $\mathbb{P}^3$ for every line in the hypersurface. It follows that for a smooth hypersurface there are 27 lines in the image, and the degree in $\mathbb{P}^3$ is 27. The use of lines residual to conics can answer many other questions too. $\endgroup$
    – Yosemite Stan
    Commented Oct 8, 2020 at 19:55

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