11
$\begingroup$

I am in this, rather friendly, situation:

I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible polynomial $P$ of degree $d$.

What i want is to get information about the existence or not of linear subvarieties of $S$, and their maximal dimension $m$. I seem to remember the existence of some ways to get bounds on $m$, given $n$ and $d$, but i don't remember anymore and i don't know where to look..

Improve this question
$\endgroup$

2 Answers 2

Reset to default
14
$\begingroup$

I think the key word you are looking for is Fano schemes of $S$. See this note by Jason Starr for a reference. For example, if $S$ is smooth and non-degenerate, it can not contain a linear subspace of dimension bigger than half $dim \ S$.

A very interesting related question is when can you find subvarieties of $S$ which was spanned in the Chow group by images of linear subspaces in $\mathbb P^n$. It was discussed in this paper by Levine, Esnault and Viehweg.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you very much, that's exactly what i was looking for! $\endgroup$
    – CuriousUser
    Feb 12, 2010 at 16:06
9
$\begingroup$

I agree with Hailong, you want to look at Fano schemes. Here's an article by Debarre and Manivel. It's very good, and I've gotten a lot out of it despite note knowing much French.

Improve this answer
$\endgroup$
1
  • $\begingroup$ That one looks great +1! $\endgroup$
    – Hailong Dao
    Feb 12, 2010 at 15:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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