\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 $\Gr(3,6)$

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 16 } &= 1 \\ \mathrm{b}_{ 18 } &= 1 \end{align*}
Basic information
dimension
9
index
6
Euler characteristic
20
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$
$\mathrm{Aut}^0(\Gr(3,6))$
$\mathrm{PGL}_{ 6 }$
$\pi_0\mathrm{Aut}(\Gr(3,6))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\Gr(3,6))$
35
Projective geometry
minimal embedding

$\Gr(3,6)\hookrightarrow\mathbb{P}^{ 19 }$

degree
42
Hilbert series
1, 20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, ...
Projective duality

The dual is the hypersurface $Q=0$ in $\mathbb P(V^\vee)\cong\mathbb P^{19}$, where $V$ is the 20-dimensional representation $\bigwedge^3\mathbb C^6$. The polynomial $Q$ is the unique, up to scale, $\mathrm{SL}_6$-invariant quartic. The invariant symplectic form on $V$ identifies this dual with the tangential variety of $\Gr(3,6)$: the union of its embedded projective tangent spaces.

codimension
1
degree
4
self-dual
no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Landsberg–Manivel, Propositions 5.4, 5.5 and 5.8 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\Gr(3,6))$:

Quantum cohomology
small
generically semisimple [Chaput–Manivel–Perrin]

Minuscule/cominuscule case, including quadric realisations.

big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\Gr(3,6))$:

Homological projective duality

Full nonrectangular Lefschetz collection. Not minimal. [Kapranov]

Lefschetz block sizes: 10, 6, 3, 1

Block sizes $(10, 6, 3, 1)$; 20 objects.

Kapranov's Schur bundles, grouped by the last row of the Young diagram (determinant twists).

No HP-dual model recorded.

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \Gr(3,6) })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), 479–508. MR939472.
  2. Pierre-Emmanuel Chaput; Laurent Manivel; Nicolas Perrin. Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences. Canad. J. Math. 62 (2010), 1246–1263. arXiv:0710.1224.
  3. Carolina Araujo; Roya Beheshti; Ana-Maria Castravet; Kelly Jabbusch; Svetlana Makarova; Enrica Mazzon; Libby Taylor; Nivedita Viswanathan. Higher Fano Manifolds. Rev. Un. Mat. Argentina 64 (2022), 103–125. arXiv:2110.02339.
  4. Alexander Samokhin; Wilberd van der Kallen. Highest weight category structures on $\operatorname{rep}(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$. Preprint (2024). arXiv:2407.13653.
  5. Changzheng Li; Zhaoyang Liu; Tony Yue Yu; Chi Zhang. Convergence and semisimplicity of big quantum cohomology of flag varieties. Preprint (2026). arXiv:2610.28418.

All references.