\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 $\mathrm{Q}^{3}$

Y Z Y Z
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 } &= 1 \end{align*}
Basic information
dimension
3
index
3
Euler characteristic
4
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$
$\mathrm{Aut}^0(\mathrm{Q}^{3})$
$\mathrm{SO}_{ 5 }$
$\pi_0\mathrm{Aut}(\mathrm{Q}^{3})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{Q}^{3})$
10
Projective geometry
minimal embedding

$\mathrm{Q}^{3}\hookrightarrow\mathbb{P}^{ 4 }$

degree
2
Hilbert series
1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, ...
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 $\mathbf D^{\mathrm b}(\mathrm{Q}^{3})$:

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.

Earlier proof for coadjoint varieties: [Perrin–Smirnov] .

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{Q}^{3})$:

Homological projective duality

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

Lefschetz block sizes: 2, 1, 1

Block sizes $(2, 1, 1)$; 4 objects.

Lefschetz centre: a spinor bundle and the structure sheaf.

HP dual: the double cover of the dual projective space branched over the dual quadric, for the spinor Lefschetz structure. [Kuznetsov–Perry, Theorem 1.1]

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{Q}^{3} })\cdot Z>0$ 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. Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), 155–182. MR2434094.
  3. 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.
  4. Alexander Kuznetsov; Alexander Perry. Homological projective duality for quadrics. J. Algebraic Geom. 30 (2021), 457–476. arXiv:1902.09832.
  5. 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.
  6. Anton Fonarev. Full exceptional collections on Lagrangian Grassmannians. Int. Math. Res. Not. 2022 (2), 1081–1122. arXiv:1911.08968.
  7. 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.
  8. Nicolas Perrin; Maxim Smirnov. On the big quantum cohomology of coadjoint varieties. Selecta Math. 31 (2025), article 30. arXiv:2112.12436.
  9. Lyalya Guseva; Alexander Novikov. Full exceptional collections on the symplectic isotropic Grassmannians. Advances in Mathematics (2026), article 111211. arXiv:2505.13005.
  10. 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.