\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.

Dual projective space $\mathbb{P}^{3,\vee}$

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
4
Euler characteristic
4
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{3,\vee})$
$\mathrm{PGL}_{ 4 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{3,\vee})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{3,\vee})$
15
Projective geometry
minimal embedding

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

degree
1
Hilbert series
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, ...
Projective duality

The dual is empty: no hyperplane contains a projective tangent space of the full projective space.

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

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\mathbb{P}^{3,\vee})$:

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 $\mathrm{c}_1(\mathbb{P}^{3,\vee})$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Beilinson]

Lefschetz block sizes: 1, 1, 1, 1

Block sizes $(1, 1, 1, 1)$; 4 objects.

The line bundles $\mathcal O,\ldots,\mathcal O(n)$.

HP dual: the zero category, for the full linear embedding and Beilinson decomposition. [Kuznetsov (HPD), Section 8]

$k$-Fanoness

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

References
  1. Beilinson, Alexander. Coherent sheaves on $𝐏^n$ and problems in linear algebra. Funct. Anal. Appl. 12 (1978), 214–216. MR509388.
  2. Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), 479–508. MR939472.
  3. Alexander Kuznetsov. Homological Projective Duality. Publ. Math. IHÉS 105 (2007), 157–220. arXiv:math/0507292.
  4. Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), 155–182. MR2434094.
  5. 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.
  6. 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.
  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. Lyalya Guseva; Alexander Novikov. Full exceptional collections on the symplectic isotropic Grassmannians. Advances in Mathematics (2026), article 111211. arXiv:2505.13005.
  9. 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.