TITLE: Varieties cut by quadrics
QUESTION [25 upvotes]: Is there a characterization of the class of varieties which can be described as an intersection of quadrics, apart from the taulogical one?
Lots of varieties arise in this way (my favorite examples are the Grassmanianns and Schubert varieties and some toric varieties) and I wonder how far can one go.

REPLY [4 votes]: Let $X\subset\mathbb{P}^N$ be a quadratic smooth variety of dimension $n$ and condimension $c$. Then:

if $n\geq c$ then $X$ is Fano.
If $n\geq c+1$ the $X$ is covered by lines. Let $x$ be a general point of $X$ and let $\mathcal{L}_x\subset\mathbb{P}^{n-1} = \mathbb{P}(T_xX)$ be the variety parametrizing lines in $X$ passing through $x$. Then $\mathcal{L}_x\subset\mathbb{P}^{n-1}$ is scheme theoretically defined by $c$ idependent quadratic equations.
If $n\geq c+2$ then: $X\subset\mathbb{P}^N$ is a complete intersection $\Leftrightarrow$ $\mathcal{L}_x\subset\mathbb{P}^{n-1}$ is a complete intersection $\Leftrightarrow$ $dim(\mathcal{L}_x)=n-c-1$.
(Hartshorne's conjecture) If $n\geq 2c+1$ then $X$ is a scheme theoretical complete intersection.
If $X$ is prime Fano (i.e. $X$ is Fano and $Pic(X) = \mathbb{Z}\left\langle H\right\rangle$) and the Fano index satifies $i(X)\geq\frac{2n+5}{3}$, then $X$ is a complete intersection.

This has been a breakthrough in the direction of Hartshorne's conjecture on complete intersections. Before it was known just for very special varieties and for Fano varieties of codimension two. You can find all of this in this paper: http://arxiv.org/abs/1209.2047.