8

How to prove that smooth cubic surface $X$ in $\mathbb{CP}^4$ is covered by lines and the normal bundle of the generic line $l$ is $N_{l/X}=\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$?

$\textbf{Edit}$

Let me give some explanations.

Basically, this question should follow from Proposition 2.13 on page 48 in Debarre's book "Higher-Dimensional Algebraic Geometry", which states that if $X$ is a subvariety in $\mathbb{P}^N$ defined by equations of degrees $d_1$,...,$d_k$ and $d:=d_1+...+d_k\leq N-1$ then through any point of $X$, there is a line contained in $X$.

Moreover, if the equations defining $X$ are general and $l$ is a general line contained in $X$ then $$N_{l/X}=\mathcal{O}_l(1)^{N-1-d}\oplus\mathcal{O}_l^{d-k}.$$

But is it true that a smooth cubic surface in $\mathbb{CP}^4$ is a complete intersection? If it is not, is there a possibility to apply this result from Debarre's book?

$\textbf{Edit 2}$

It seems, however, that I can not apply this result from Debarre's book. If my cubic surface is given by two homogeneous equations, the first one of degree 1 and the second one of degree 3, then $d=4$ and the conditions of the Proposition are not satisfied.

Could anyone suggest me how to think about this problem?

guest31
  • 925
  • 5
    This is a perfectly formulated, non-trivial and interesting question, at least for anyone with a modest competence in algebraic geometry. I have no idea why the users who put the question on hold think otherwise. I vote to reopen and urge other users to do the same and upvote this fine question. – Georges Elencwajg Dec 02 '15 at 10:31
  • I made an edit. – guest31 Dec 02 '15 at 20:52
  • 2
    As the normal bundle is supposed to be 2-dimensional, X is probably not a cubic surface, but a cubic hypersurface of dimension 3, then the proposition applies. – evgeny Dec 03 '15 at 15:13

0 Answers0