\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 $\mathrm{Q}^{6}$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 12 } &= 1 \end{align*}
Basic information
dimension
6
index
6
Euler characteristic
8
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{6})$
$\mathrm{PSO}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{6})$
$\mathbb{Z}/3\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{6})$
28
Projective geometry
minimal embedding

$\mathrm{Q}^{6}\hookrightarrow\mathbb{P}^{ 7 }$

degree
2
Hilbert series
1, 8, 35, 112, 294, 672, 1386, 2640, 4719, 8008, 13013, 20384, 30940, 45696, 65892, 93024, 128877, 175560, 235543, 311696, ...
Projective duality

A smooth quadric in the dual projective space.

codimension
1
degree
2
self-dual
yes

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] All Grassmannians.

Exceptional collections

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

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 $\mathrm{c}_1(\mathrm{Q}^{6})$:

Homological projective duality

Full minimal nonrectangular Lefschetz collection. [Kuznetsov–Perry, Lemma 2.4]

Lefschetz block sizes: 2, 2, 1, 1, 1, 1

Block sizes $(2, 2, 1, 1, 1, 1)$; 8 objects.

Lefschetz centre: a spinor bundle and the structure sheaf.

HP dual: the classical dual quadric, for the spinor Lefschetz structure. [Kuznetsov–Perry, Theorem 1.1]

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{Q}^{6} })\cdot Z>0$ 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. Alexander Kuznetsov; Alexander Perry. Homological projective duality for quadrics. J. Algebraic Geom. 30 (2021), 457–476. arXiv:1902.09832.
  4. 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.
  5. 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.
  6. 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.