\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.

Symplectic Grassmannian $\SGr(3,14)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 4 \\ \mathrm{b}_{ 6 } &= 5 \\ \mathrm{b}_{ 7 } &= 7 \\ \mathrm{b}_{ 8 } &= 8 \\ \mathrm{b}_{ 9 } &= 10 \\ \mathrm{b}_{ 10 } &= 12 \\ \mathrm{b}_{ 11 } &= 13 \\ \mathrm{b}_{ 12 } &= 15 \\ \mathrm{b}_{ 13 } &= 16 \\ \mathrm{b}_{ 14 } &= 17 \\ \mathrm{b}_{ 15 } &= 17 \\ \mathrm{b}_{ 16 } &= 18 \\ \mathrm{b}_{ 17 } &= 17 \\ \mathrm{b}_{ 18 } &= 17 \\ \mathrm{b}_{ 19 } &= 16 \\ \mathrm{b}_{ 20 } &= 15 \\ \mathrm{b}_{ 21 } &= 13 \\ \mathrm{b}_{ 22 } &= 12 \\ \mathrm{b}_{ 23 } &= 10 \\ \mathrm{b}_{ 24 } &= 8 \\ \mathrm{b}_{ 25 } &= 7 \\ \mathrm{b}_{ 26 } &= 5 \\ \mathrm{b}_{ 27 } &= 4 \\ \mathrm{b}_{ 28 } &= 3 \\ \mathrm{b}_{ 29 } &= 2 \\ \mathrm{b}_{ 30 } &= 1 \\ \mathrm{b}_{ 31 } &= 1 \end{align*}
Basic information
dimension
30
index
12
Euler characteristic
280
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 4$, $\mathrm{b}_{ 6 } = 5$, $\mathrm{b}_{ 7 } = 7$, $\mathrm{b}_{ 8 } = 8$, $\mathrm{b}_{ 9 } = 10$, $\mathrm{b}_{ 10 } = 12$, $\mathrm{b}_{ 11 } = 13$, $\mathrm{b}_{ 12 } = 15$, $\mathrm{b}_{ 13 } = 16$, $\mathrm{b}_{ 14 } = 17$, $\mathrm{b}_{ 15 } = 17$, $\mathrm{b}_{ 16 } = 18$, $\mathrm{b}_{ 17 } = 17$, $\mathrm{b}_{ 18 } = 17$, $\mathrm{b}_{ 19 } = 16$, $\mathrm{b}_{ 20 } = 15$, $\mathrm{b}_{ 21 } = 13$, $\mathrm{b}_{ 22 } = 12$, $\mathrm{b}_{ 23 } = 10$, $\mathrm{b}_{ 24 } = 8$, $\mathrm{b}_{ 25 } = 7$, $\mathrm{b}_{ 26 } = 5$, $\mathrm{b}_{ 27 } = 4$, $\mathrm{b}_{ 28 } = 3$, $\mathrm{b}_{ 29 } = 2$, $\mathrm{b}_{ 30 } = 1$, $\mathrm{b}_{ 31 } = 1$
$\mathrm{Aut}^0(\SGr(3,14))$
$\mathrm{PSp}_{ 14 }$
$\pi_0\mathrm{Aut}(\SGr(3,14))$
$1$
$\dim\mathrm{Aut}^0(\SGr(3,14))$
105
Projective geometry
minimal embedding

$\SGr(3,14)\hookrightarrow\mathbb{P}^{ 349 }$

degree
47052314400
Hilbert series
1, 350, 37400, 1939938, 60575515, 1290577288, 20352332000, 251433468000, 2536344778968, 21557100646640, 158184517848448, 1021771893360400, 5901772949658600, 30878718442458656, 147930458703711200, 654809824556133400, 2698876063882659375, 10426477615910606250, 37972420263785007000, 131021858832908793750, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\SGr(3,14))$. Will you be the first to construct one? Let us know if you do!

Kuznetsov–Polishchuk have constructed an exceptional collection of maximal length in MR3463417. Can you prove it's full?