\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 $\Gr(3,5)$

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 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 12 } &= 1 \end{align*}
Basic information
dimension
6
index
5
Euler characteristic
10
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$
$\mathrm{Aut}^0(\Gr(3,5))$
$\mathrm{PGL}_{ 5 }$
$\pi_0\mathrm{Aut}(\Gr(3,5))$
$1$
$\dim\mathrm{Aut}^0(\Gr(3,5))$
24
Projective geometry
minimal embedding

$\Gr(3,5)\hookrightarrow\mathbb{P}^{ 9 }$

degree
5
Hilbert series
1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, ...
Projective duality

The Pfaffian locus of skew forms of rank at most 2.

codimension
3
degree
5
self-dual
yes

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

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\Gr(3,5))$:

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 $\mathrm{c}_1(\Gr(3,5))$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Kuznetsov, Theorem 4.1]

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

Block sizes $(2, 2, 2, 2, 2)$; 10 objects.

Symmetric powers of the dual tautological bundle form the first block.

HP dual: the Grassmannian of two-planes in the dual five-dimensional vector space. [Kuznetsov, Section 6.1]

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \Gr(3,5) })\cdot Z>0$ 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, A. G. Hyperplane sections and derived categories. Izv. Math. 70 (2006), 447–547. MR2238172.
  3. Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), 155–182. MR2434094.
  4. 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.
  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. 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.
  7. 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.