\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.

Projective space $\mathbb{P}^{3}$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \end{align*}
Basic information
dimension
3
index
4
Euler characteristic
4
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{3})$
$\mathrm{PGL}_{ 4 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{3})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{3})$
15
Projective geometry
minimal embedding

$\mathbb{P}^{3}\hookrightarrow\mathbb{P}^{ 3 }$

More appropriately in this particular presentation as a quotient of the symplectic group is to consider the Grassmannian as the adjoint variety of type $\mathrm{C}_{ 2 }$, where the embedding is the second Veronese embedding into $\mathbb{P}(\mathrm{V}_{2\omega_1})=\mathbb{P}^{ 9 }$.

degree
1
Hilbert series
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, ...
Exceptional collections
  • Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathbb{P}^{3})$ are given by:

Homological projective duality