# Can projective hypersurfaces contain linear spaces? How big?

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..

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## 2 Answers

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.

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Thank you very much, that's exactly what i was looking for! –  Marco Radeschi Feb 12 '10 at 16:06

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.

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That one looks great +1! –  Hailong Dao Feb 12 '10 at 15:35