\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.

Grassmannian $\Gr(4,10)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 5 \\ \mathrm{b}_{ 10 } &= 6 \\ \mathrm{b}_{ 12 } &= 9 \\ \mathrm{b}_{ 14 } &= 10 \\ \mathrm{b}_{ 16 } &= 13 \\ \mathrm{b}_{ 18 } &= 14 \\ \mathrm{b}_{ 20 } &= 16 \\ \mathrm{b}_{ 22 } &= 16 \\ \mathrm{b}_{ 24 } &= 18 \\ \mathrm{b}_{ 26 } &= 16 \\ \mathrm{b}_{ 28 } &= 16 \\ \mathrm{b}_{ 30 } &= 14 \\ \mathrm{b}_{ 32 } &= 13 \\ \mathrm{b}_{ 34 } &= 10 \\ \mathrm{b}_{ 36 } &= 9 \\ \mathrm{b}_{ 38 } &= 6 \\ \mathrm{b}_{ 40 } &= 5 \\ \mathrm{b}_{ 42 } &= 3 \\ \mathrm{b}_{ 44 } &= 2 \\ \mathrm{b}_{ 46 } &= 1 \\ \mathrm{b}_{ 48 } &= 1 \end{align*}
Basic information
dimension
24
index
10
Euler characteristic
210
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 9$, $\mathrm{b}_{ 14 } = 10$, $\mathrm{b}_{ 16 } = 13$, $\mathrm{b}_{ 18 } = 14$, $\mathrm{b}_{ 20 } = 16$, $\mathrm{b}_{ 22 } = 16$, $\mathrm{b}_{ 24 } = 18$, $\mathrm{b}_{ 26 } = 16$, $\mathrm{b}_{ 28 } = 16$, $\mathrm{b}_{ 30 } = 14$, $\mathrm{b}_{ 32 } = 13$, $\mathrm{b}_{ 34 } = 10$, $\mathrm{b}_{ 36 } = 9$, $\mathrm{b}_{ 38 } = 6$, $\mathrm{b}_{ 40 } = 5$, $\mathrm{b}_{ 42 } = 3$, $\mathrm{b}_{ 44 } = 2$, $\mathrm{b}_{ 46 } = 1$, $\mathrm{b}_{ 48 } = 1$
$\mathrm{Aut}^0(\Gr(4,10))$
$\mathrm{PGL}_{ 10 }$
$\pi_0\mathrm{Aut}(\Gr(4,10))$
$1$
$\dim\mathrm{Aut}^0(\Gr(4,10))$
99
Projective geometry
minimal embedding

$\Gr(4,10)\hookrightarrow\mathbb{P}^{ 209 }$

degree
140229804
Hilbert series
1, 210, 13860, 457380, 9343620, 133613766, 1447482465, 12544848030, 90474964580, 559299781040, 3031952379456, 14675134144320, 64344818940480, 258616676126160, 962206162645860, 3341308066756506, 10904939897954025, 33648885753232750, 98669664990682500, 276180641241862500, ...
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.

Homological projective duality