\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

Quadric $\mathrm{Q}^{11}$

Y Z Y Z Y Z Y Z Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 14 } &= 1 \\ \mathrm{b}_{ 16 } &= 1 \\ \mathrm{b}_{ 18 } &= 1 \\ \mathrm{b}_{ 20 } &= 1 \\ \mathrm{b}_{ 22 } &= 1 \end{align*}
Basic information
dimension
11
index
11
Euler characteristic
12
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{11})$
$\mathrm{SO}_{ 13 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{11})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{11})$
78
Projective geometry
minimal embedding

$\mathrm{Q}^{11}\hookrightarrow\mathbb{P}^{ 12 }$

degree
2
Hilbert series
1, 13, 90, 442, 1729, 5733, 16744, 44200, 107406, 243542, 520676, 1058148, 2057510, 3848222, 6953544, 12183560, 20764055, 34512075, 56071470, 89224590, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{Q}^{11})$ are given by:

Homological projective duality