SGr(3,16)—grassmannian.info

Betti numbers

\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 10 \\ \mathrm{b}_{ 18 } &= 12 \\ \mathrm{b}_{ 20 } &= 14 \\ \mathrm{b}_{ 22 } &= 16 \\ \mathrm{b}_{ 24 } &= 18 \\ \mathrm{b}_{ 26 } &= 20 \\ \mathrm{b}_{ 28 } &= 21 \\ \mathrm{b}_{ 30 } &= 23 \\ \mathrm{b}_{ 32 } &= 23 \\ \mathrm{b}_{ 34 } &= 24 \\ \mathrm{b}_{ 36 } &= 24 \\ \mathrm{b}_{ 38 } &= 24 \\ \mathrm{b}_{ 40 } &= 23 \\ \mathrm{b}_{ 42 } &= 23 \\ \mathrm{b}_{ 44 } &= 21 \\ \mathrm{b}_{ 46 } &= 20 \\ \mathrm{b}_{ 48 } &= 18 \\ \mathrm{b}_{ 50 } &= 16 \\ \mathrm{b}_{ 52 } &= 14 \\ \mathrm{b}_{ 54 } &= 12 \\ \mathrm{b}_{ 56 } &= 10 \\ \mathrm{b}_{ 58 } &= 8 \\ \mathrm{b}_{ 60 } &= 7 \\ \mathrm{b}_{ 62 } &= 5 \\ \mathrm{b}_{ 64 } &= 4 \\ \mathrm{b}_{ 66 } &= 3 \\ \mathrm{b}_{ 68 } &= 2 \\ \mathrm{b}_{ 70 } &= 1 \\ \mathrm{b}_{ 72 } &= 1 \end{align*}

Basic information

dimension: 36
index: 14
Euler characteristic: 448
Betti numbers: $\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 10$, $\mathrm{b}_{ 18 } = 12$, $\mathrm{b}_{ 20 } = 14$, $\mathrm{b}_{ 22 } = 16$, $\mathrm{b}_{ 24 } = 18$, $\mathrm{b}_{ 26 } = 20$, $\mathrm{b}_{ 28 } = 21$, $\mathrm{b}_{ 30 } = 23$, $\mathrm{b}_{ 32 } = 23$, $\mathrm{b}_{ 34 } = 24$, $\mathrm{b}_{ 36 } = 24$, $\mathrm{b}_{ 38 } = 24$, $\mathrm{b}_{ 40 } = 23$, $\mathrm{b}_{ 42 } = 23$, $\mathrm{b}_{ 44 } = 21$, $\mathrm{b}_{ 46 } = 20$, $\mathrm{b}_{ 48 } = 18$, $\mathrm{b}_{ 50 } = 16$, $\mathrm{b}_{ 52 } = 14$, $\mathrm{b}_{ 54 } = 12$, $\mathrm{b}_{ 56 } = 10$, $\mathrm{b}_{ 58 } = 8$, $\mathrm{b}_{ 60 } = 7$, $\mathrm{b}_{ 62 } = 5$, $\mathrm{b}_{ 64 } = 4$, $\mathrm{b}_{ 66 } = 3$, $\mathrm{b}_{ 68 } = 2$, $\mathrm{b}_{ 70 } = 1$, $\mathrm{b}_{ 72 } = 1$
$\mathrm{Aut}^0(\SGr(3,16))$: $\mathrm{PSp}_{ 16 }$
$\pi_0\mathrm{Aut}(\SGr(3,16))$: $1$
$\dim\mathrm{Aut}^0(\SGr(3,16))$: 136

Projective geometry

minimal embedding: $\SGr(3,16)\hookrightarrow\mathbb{P}^{ 543 }$
degree: 18311313000960
Hilbert series: 1, 544, 88179, 6777680, 306850544, 9290910720, 204452333280, 3466253404224, 47259797571480, 535393752717824, 5170165780403624, 43437048909487296, 322822965534795040, 2151715762963072000, 13011156254167326000, 72074712047833638000, 368804199631095876375, 1755759739749275988000, 7825120177359843028125, 32827225654646766252000, ...

Projective duality

The projective dual is a hypersurface.

codimension: 1
degree: 188032096
self-dual: no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Tevelev, Theorems 7.1–7.2 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\SGr(3,16))$:

Samokhin–van der Kallen (2024)
Novikov (2025)
Guseva–Novikov (2026)

Quantum cohomology

small: semisimplicity unknown
big: generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]
Convergent near the large-radius limit.

Homological projective duality

Full minimal rectangular Lefschetz collection. [Novikov, Theorem 1 and Remark 2]

Block sizes $(32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32)$; 448 objects.

Equivariant bundles built from Schur functors of the tautological bundle, symplectic exterior powers, and staircase complexes.

No HP-dual model recorded.

$k$-Fanoness

Fano but not 2-Fano. [Araujo et al., Theorems 1.3 and 1.4]

References

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.
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.
Novikov, A. A. Minimal Lefschetz collections on isotropic Grassmannians $IGr(3,2n)$. Russian Math. Surveys 80 (2025), 543–545. MR4961293.
Lyalya Guseva; Alexander Novikov. Full exceptional collections on the symplectic isotropic Grassmannians. Advances in Mathematics (2026), article 111211. arXiv:2505.13005.
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.

Grassmannian.info

Symplectic Grassmannian $\SGr(3,16)$