0
$\begingroup$

Let $C$ be a smooth complete intersection of two smooth hypersurfaces of degree atleast $4$ in $\mathbb P^3$. Does it necessarily mean that it can not be a smooth plane curve?

If not in general, then can we do the following : Let $X$ be a general surface of degree $d_1 \geq 4$ in $\mathbb P^3$. Let $H$ be the hyperplane divisor. Then can we always choose $C \in |d_2H|$ ($d_2 \geq 4$) such that $C$ becomes a smooth complete intersection of surfaces of degree $d_1,d_2$ in $\mathbb P^3$ but $C$ is not contained in any hyperpalne?

Does some argument along the lines of Bertinis theorem help?

$\endgroup$
2
  • 1
    $\begingroup$ It is not clear from the way you formulated the question whether the curve is of degree at least 4 or the surfaces are of degree at least 4. In the former case, why can you not have $C=P\cap Q$ where $P$ is a plane and $Q$ a surface of degre at least 4? $\endgroup$ Commented Sep 17, 2022 at 4:23
  • $\begingroup$ @Kapil, I meant both the surfaces are of degree at least $4$. $\endgroup$ Commented Sep 17, 2022 at 4:34

1 Answer 1

Reset to default
3
$\begingroup$

Using the Koszul resolution $$ 0 \to \mathcal{O}(-d_1-d_2) \to \mathcal{O}(-d_1) \oplus \mathcal{O}(-d_2) \to I_C \to 0 $$ it is easy to check that $H^0(\mathbb{P}^3, I_C(1)) = 0$, hence $C$ is not contained in a hyperplane.

$\endgroup$
8
  • $\begingroup$ thank you very much for the answer. Could you please give a reference how this short exact sequence is constructed? In this context are we using the fact that $C$ is a strict complete intersection? Is it trivial that we can always choose $C \in |d_2H|$ on $X$ such that it is smooth and moreover it is a strict complete intersection of surfaces of degrees $d_1,d_2$ (and not just a set theoretic complete intersection)? $\endgroup$ Commented Sep 17, 2022 at 9:25
  • 2
    $\begingroup$ The original question was whether the curve can be a plane curve. I took this to mean that there exists a (different) line bundle on the curve such that it gives an embedding to $\mathbb{P}^2$. With the genericity hypothesis, this ought to be true. (For example, if one surface is of degree $4$, then results of Ein-Lazarsfeld say that there are no (other) special line bundles on the curve.) However, proving this would require more work on existence of linear systems on special families of curves. $\endgroup$ Commented Sep 17, 2022 at 10:38
  • 1
    $\begingroup$ @New: This is called Koszul complex, I think you can find it in any textbook, e.g., it should be in Hartshorne where he discusses regular sequences. Yes, it is important that $C$ is a complete intersection as a scheme. And yes, a general complete intersection is smooth by Bertini. $\endgroup$ Commented Sep 17, 2022 at 16:35
  • $\begingroup$ @Kapil: I agree, the question about realization of a complete intersection curve as a plane curve is much more subtle. $\endgroup$ Commented Sep 17, 2022 at 16:36
  • 1
    $\begingroup$ @New: Any such $C$ is complete intersection. $\endgroup$ Commented Sep 17, 2022 at 17:54

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.