\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 $\mathrm{Q}^{12}$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 5 } &= 1 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 7 } &= 2 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 9 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 11 } &= 1 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 13 } &= 1 \end{align*}
Basic information
dimension
12
index
12
Euler characteristic
14
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 2$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 11 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 13 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{12})$
$\mathrm{PSO}_{ 14 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{12})$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{12})$
91
Projective geometry
minimal embedding

$\mathrm{Q}^{12}\hookrightarrow\mathbb{P}^{ 13 }$

degree
2
Hilbert series
1, 14, 104, 546, 2275, 8008, 24752, 68952, 176358, 419900, 940576, 1998724, 4056234, 7904456, 14858000, 27041560, 47805615, 82317690, 138389160, 227613750, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.