ag.algebraic geometry - Varieties cut by quadrics

As Pete already indicated, Mumford's theorem says that for any projective variety $Xsubset mathbb P^n$, its Veronese emberdding $v_d(X)subset mathbb P^N$ is cut out by quadrics, for $dgg0$. So a reasonable question is for the variety with a fixed projective embedding (such as Grassmannians in the Plucker embedding and not in some other random embedding).

For this latter more meaningful question, many "combinatorial" rational varieties, such as Grassmannians, Schubert varieties (as you pointed out), flag varieties, determinantal varieties, etc., are cut out by quadrics.

For the "non-combinatorial", non-rational varieties, the most classical result is Petri's theorem: a smooth non-hyperelliptic curve of genus $gge 4$ in its canonical embedding is cut out by quadrics, with the exceptions of trigonal curves and plane quintics.

There is a vast generatization of this property: $Xsubset mathbb P^n$ satisfies property $N_p$ if the first syzygy of its homogeneous ideal $I_X$ is a direct sum of $mathcal O(2)$, the second syzygy is a direct sum of $mathcal O(3)$, etc., the $p$-th syzygy is a direct sum of $mathcal O(p+1)$. In this language, $X$ is cut out by quadrics is equivalent to the property $N_1$.

A 1984 Green's conjecture is that a smooth nonhyperelliptic curve satisfies $N_p$ for $p=Cliff(X)-1$, where $Cliff(X)$ is the Clifford of $X$. This has been proved for generic curves of any genus by Voisin (in characteristic 0; it is false in positive characteristic).

Another notable case: the ideal of $2times 2$ minors of a $ptimes q$ matrix has property $N_{p+q-3}$.

• Next co.combinatorics - efficient way to count hamiltonian paths in a grid graph for a given pair of vertices
• Previous lo.logic - What is the manner of inconsistency of Girard's paradox in Martin Lof type theory