\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 $\mathrm{Q}^{13}$

Y Z 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 \\ \mathrm{b}_{ 24 } &= 1 \\ \mathrm{b}_{ 26 } &= 1 \end{align*}
Basic information
dimension
13
index
13
Euler characteristic
14
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 22 } = 1$, $\mathrm{b}_{ 24 } = 1$, $\mathrm{b}_{ 26 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{13})$
$\mathrm{SO}_{ 15 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{13})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{13})$
105
Projective geometry
minimal embedding

$\mathrm{Q}^{13}\hookrightarrow\mathbb{P}^{ 14 }$

degree
2
Hilbert series
1, 15, 119, 665, 2940, 10948, 35700, 104652, 281010, 700910, 1641486, 3640210, 7696444, 15600900, 30458900, 57500460, 105306075, 187623765, 326012925, 553626675, ...
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 $\mathrm{c}_1(\mathrm{Q}^{13})$ are given by:

Homological projective duality