Projective Spaces¶
Index convention: the displayed Ambient Index is the software's ambient
Chern gcd. In particular, P2 / O(2) has displayed index 1 but is a smooth
conic, hence P1, with geometric Fano index 2. See the
catalog conventions.
This section contains complete intersections inside this geometric family, systematically organized by dimension and ambient index.
For the kept-root beta convention and copyable browser/CLI syntax, see How to Use.
Reproduce an entry in the software¶
Every card below is a complete intersection in projective space. For the
ambient space $P^n$, run A(n) with Kept simple roots 1. Each
displayed summand / O(d) is one row in Bundle K: enter d for one
summand, or d1;d2;... for several summands. For example, the card
P8 / O(2)+O(3) is reproduced with A8, kept roots 1, and Bundle K
2;3 (or [[2],[3]] in Python). The card's dimension is n minus the
number of rows, and its ambient index is n+1 - sum(d); these provide quick
checks that the input matches the card.
8-folds¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P8 Fano index 9 | 9 | 9 |
P8 Fano index 9
- Ambient Space:
$A8$ - Dimension: \(8\)
- Ambient Index: \(9\)
- Basis Rank: \(9\)
- Eigenvalues: \(9\) (mult: 1), \(- 9 \cos{\left(\frac{\pi}{9} \right)} - 9 i \sin{\left(\frac{\pi}{9} \right)}\) (mult: 1), \(- 9 \cos{\left(\frac{\pi}{9} \right)} + 9 i \sin{\left(\frac{\pi}{9} \right)}\) (mult: 1), \(- \frac{9}{2} - \frac{9 \sqrt{3} i}{2}\) (mult: 1), \(- \frac{9}{2} + \frac{9 \sqrt{3} i}{2}\) (mult: 1), \(9 \cos{\left(\frac{4 \pi}{9} \right)} - 9 i \sin{\left(\frac{4 \pi}{9} \right)}\) (mult: 1), \(9 \cos{\left(\frac{4 \pi}{9} \right)} + 9 i \sin{\left(\frac{4 \pi}{9} \right)}\) (mult: 1), \(9 \cos{\left(\frac{2 \pi}{9} \right)} - 9 i \sin{\left(\frac{2 \pi}{9} \right)}\) (mult: 1), \(9 \cos{\left(\frac{2 \pi}{9} \right)} + 9 i \sin{\left(\frac{2 \pi}{9} \right)}\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9\\9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0\end{bmatrix}
\]
7-folds¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P7 Fano index 8 | 8 | 8 |
| P8 / O(2) | 7 | 8 |
| P8 / O(3) | 6 | 8 |
| P8 / O(4) | 5 | 8 |
| P8 / O(5) | 4 | 8 |
| P8 / O(6) | 3 | 8 |
| P8 / O(7) | 2 | 8 |
P7 Fano index 8
- Ambient Space:
$A7$ - Dimension: \(7\)
- Ambient Index: \(8\)
- Basis Rank: \(8\)
- Eigenvalues: \(-8\) (mult: 1), \(8\) (mult: 1), \(- 4 \sqrt{2} - 4 \sqrt{2} i\) (mult: 1), \(- 4 \sqrt{2} + 4 \sqrt{2} i\) (mult: 1), \(- 8 i\) (mult: 1), \(8 i\) (mult: 1), \(4 \sqrt{2} - 4 \sqrt{2} i\) (mult: 1), \(4 \sqrt{2} + 4 \sqrt{2} i\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 8\\8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 8 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 8 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 8 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\end{bmatrix}
\]
P8 / O(2)
- Ambient Space:
$A8$ - Dimension: \(7\)
- Ambient Index: \(7\)
- Basis Rank: \(8\)
- Eigenvalues: \(7 \cdot 2^{\frac{2}{7}}\) (mult: 1), \(- 7 \cdot 2^{\frac{2}{7}} \cos{\left(\frac{\pi}{7} \right)} - 7 \cdot 2^{\frac{2}{7}} i \sin{\left(\frac{\pi}{7} \right)}\) (mult: 1), \(- 7 \cdot 2^{\frac{2}{7}} \cos{\left(\frac{\pi}{7} \right)} + 7 \cdot 2^{\frac{2}{7}} i \sin{\left(\frac{\pi}{7} \right)}\) (mult: 1), \(- 7 \cdot 2^{\frac{2}{7}} \cos{\left(\frac{3 \pi}{7} \right)} - 7 \cdot 2^{\frac{2}{7}} i \sin{\left(\frac{3 \pi}{7} \right)}\) (mult: 1), \(- 7 \cdot 2^{\frac{2}{7}} \cos{\left(\frac{3 \pi}{7} \right)} + 7 \cdot 2^{\frac{2}{7}} i \sin{\left(\frac{3 \pi}{7} \right)}\) (mult: 1), \(7 \cdot 2^{\frac{2}{7}} \cos{\left(\frac{2 \pi}{7} \right)} - 7 \cdot 2^{\frac{2}{7}} i \sin{\left(\frac{2 \pi}{7} \right)}\) (mult: 1), \(7 \cdot 2^{\frac{2}{7}} \cos{\left(\frac{2 \pi}{7} \right)} + 7 \cdot 2^{\frac{2}{7}} i \sin{\left(\frac{2 \pi}{7} \right)}\) (mult: 1), \(0\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 14 & 0\\7 & 0 & 0 & 0 & 0 & 0 & 0 & 14\\0 & 7 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 7 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 7 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 7 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 7 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 7 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & & 1 & & & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & & & 1 & & & & & & & \\
\end{matrix}
\]
P8 / O(3)
- Ambient Space:
$A8$ - Dimension: \(7\)
- Ambient Index: \(6\)
- Basis Rank: \(8\)
- Eigenvalues: \(- 6 \sqrt{3}\) (mult: 1), \(6 \sqrt{3}\) (mult: 1), \(- 3 \sqrt{3} - 9 i\) (mult: 1), \(- 3 \sqrt{3} + 9 i\) (mult: 1), \(3 \sqrt{3} - 9 i\) (mult: 1), \(3 \sqrt{3} + 9 i\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 36 & 0 & 0\\6 & 0 & 0 & 0 & 0 & 0 & 90 & 0\\0 & 6 & 0 & 0 & 0 & 0 & 0 & 36\\0 & 0 & 6 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 6 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 6 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 6 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & & 1 & & & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 1 & & 84 & & 84 & & 1 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & & & 1 & & & & & & & \\
\end{matrix}
\]
P8 / O(4)
- Ambient Space:
$A8$ - Dimension: \(7\)
- Ambient Index: \(5\)
- Basis Rank: \(8\)
- Eigenvalues: \(10 \cdot 2^{\frac{3}{5}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{3}{5}} \sqrt{5}}{2} - \frac{5 \cdot 2^{\frac{3}{5}}}{2} - 10 \cdot 2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{3}{5}} \sqrt{5}}{2} - \frac{5 \cdot 2^{\frac{3}{5}}}{2} + 10 \cdot 2^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{3}{5}}}{2} + \frac{5 \cdot 2^{\frac{3}{5}} \sqrt{5}}{2} - 10 \cdot 2^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{3}{5}}}{2} + \frac{5 \cdot 2^{\frac{3}{5}} \sqrt{5}}{2} + 10 \cdot 2^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 120 & 0 & 0 & 0\\5 & 0 & 0 & 0 & 0 & 520 & 0 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 520 & 0\\0 & 0 & 5 & 0 & 0 & 0 & 0 & 120\\0 & 0 & 0 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 5 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & & 1 & & & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 156 & & 2304 & & 2304 & & 156 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & & & 1 & & & & & & & \\
\end{matrix}
\]
P8 / O(5)
- Ambient Space:
$A8$ - Dimension: \(7\)
- Ambient Index: \(4\)
- Basis Rank: \(8\)
- Eigenvalues: \(- 20 \sqrt[4]{5}\) (mult: 1), \(20 \sqrt[4]{5}\) (mult: 1), \(- 20 \sqrt[4]{5} i\) (mult: 1), \(20 \sqrt[4]{5} i\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 480 & 0 & 0 & 0 & 57600\\4 & 0 & 0 & 0 & 3080 & 0 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 5380 & 0 & 0\\0 & 0 & 4 & 0 & 0 & 0 & 3080 & 0\\0 & 0 & 0 & 4 & 0 & 0 & 0 & 480\\0 & 0 & 0 & 0 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & & 1 & & & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
0 & & 9 & & 2598 & & 23607 & & 23607 & & 2598 & & 9 & & 0 \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & & & 1 & & & & & & & \\
\end{matrix}
\]
P8 / O(6)
- Ambient Space:
$A8$ - Dimension: \(7\)
- Ambient Index: \(3\)
- Basis Rank: \(8\)
- Eigenvalues: \(108\) (mult: 1), \(-54 - 54 \sqrt{3} i\) (mult: 1), \(-54 + 54 \sqrt{3} i\) (mult: 1), \(0\) (mult: 5)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 2160 & 0 & 0 & 42456960 & 0 & 0\\3 & 0 & 0 & 18792 & 0 & 0 & 132020928 & 0\\0 & 3 & 0 & 0 & 49032 & 0 & 0 & 42456960\\0 & 0 & 3 & 0 & 0 & 49032 & 0 & 0\\0 & 0 & 0 & 3 & 0 & 0 & 18792 & 0\\0 & 0 & 0 & 0 & 3 & 0 & 0 & 2160\\0 & 0 & 0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & & 1 & & & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
0 & & 165 & & 19855 & & 142740 & & 142740 & & 19855 & & 165 & & 0 \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & & & 1 & & & & & & & \\
\end{matrix}
\]
P8 / O(7)
- Ambient Space:
$A8$ - Dimension: \(7\)
- Ambient Index: \(2\)
- Basis Rank: \(8\)
- Eigenvalues: \(- 686 \sqrt{7}\) (mult: 1), \(686 \sqrt{7}\) (mult: 1), \(0\) (mult: 6)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 10080 & 0 & 4112519040 & 0 & 1068310404604800 & 0 & 10225965588972134400\\2 & 0 & 112392 & 0 & 49399013664 & 0 & 3840731271980064 & 0\\0 & 2 & 0 & 400904 & 0 & 107503371248 & 0 & 1068310404604800\\0 & 0 & 2 & 0 & 600334 & 0 & 49399013664 & 0\\0 & 0 & 0 & 2 & 0 & 400904 & 0 & 4112519040\\0 & 0 & 0 & 0 & 2 & 0 & 112392 & 0\\0 & 0 & 0 & 0 & 0 & 2 & 0 & 10080\\0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & & 1 & & & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
0 & & 1287 & & 98979 & & 619569 & & 619569 & & 98979 & & 1287 & & 0 \\
& 0 & & 0 & & 0 & & 1 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 1 & & 0 & & 0 & & & \\
& & & & 0 & & 0 & & 0 & & 0 & & & & \\
& & & & & 0 & & 1 & & 0 & & & & & \\
& & & & & & 0 & & 0 & & & & & & \\
& & & & & & & 1 & & & & & & & \\
\end{matrix}
\]
6-folds¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P6 Fano index 7 | 7 | 7 |
| P7 / O(2) | 6 | 7 |
| P7 / O(3) | 5 | 7 |
| P8 / O(2)+O(2) | 5 | 7 |
| P7 / O(4) | 4 | 7 |
| P8 / O(2)+O(3) | 4 | 7 |
| P7 / O(5) | 3 | 7 |
| P8 / O(2)+O(4) | 3 | 7 |
| P8 / O(3)+O(3) | 3 | 7 |
| P7 / O(6) | 2 | 7 |
| P8 / O(2)+O(5) | 2 | 7 |
| P8 / O(3)+O(4) | 2 | 7 |
P6 Fano index 7
- Ambient Space:
$A6$ - Dimension: \(6\)
- Ambient Index: \(7\)
- Basis Rank: \(7\)
- Eigenvalues: \(7\) (mult: 1), \(- 7 \cos{\left(\frac{\pi}{7} \right)} - 7 i \sin{\left(\frac{\pi}{7} \right)}\) (mult: 1), \(- 7 \cos{\left(\frac{\pi}{7} \right)} + 7 i \sin{\left(\frac{\pi}{7} \right)}\) (mult: 1), \(- 7 \cos{\left(\frac{3 \pi}{7} \right)} - 7 i \sin{\left(\frac{3 \pi}{7} \right)}\) (mult: 1), \(- 7 \cos{\left(\frac{3 \pi}{7} \right)} + 7 i \sin{\left(\frac{3 \pi}{7} \right)}\) (mult: 1), \(7 \cos{\left(\frac{2 \pi}{7} \right)} - 7 i \sin{\left(\frac{2 \pi}{7} \right)}\) (mult: 1), \(7 \cos{\left(\frac{2 \pi}{7} \right)} + 7 i \sin{\left(\frac{2 \pi}{7} \right)}\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 7\\7 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 7 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 7 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 7 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 7 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 7 & 0\end{bmatrix}
\]
P7 / O(2)
- Ambient Space:
$A7$ - Dimension: \(6\)
- Ambient Index: \(6\)
- Basis Rank: \(7\)
- Eigenvalues: \(- 6 \sqrt[3]{2}\) (mult: 1), \(6 \sqrt[3]{2}\) (mult: 1), \(- 3 \sqrt[3]{2} - 3 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(- 3 \sqrt[3]{2} + 3 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(3 \sqrt[3]{2} - 3 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(3 \sqrt[3]{2} + 3 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(0\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 12 & 0\\6 & 0 & 0 & 0 & 0 & 0 & 12\\0 & 6 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 6 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 6 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 6 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 0 & & 2 & & 0 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P7 / O(3)
- Ambient Space:
$A7$ - Dimension: \(6\)
- Ambient Index: \(5\)
- Basis Rank: \(7\)
- Eigenvalues: \(5 \cdot 3^{\frac{3}{5}}\) (mult: 1), \(- \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} - \frac{5 \cdot 3^{\frac{3}{5}}}{4} - 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} - \frac{5 \cdot 3^{\frac{3}{5}}}{4} + 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 3^{\frac{3}{5}}}{4} + \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} - 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(- \frac{5 \cdot 3^{\frac{3}{5}}}{4} + \frac{5 \cdot 3^{\frac{3}{5}} \sqrt{5}}{4} + 5 \cdot 3^{\frac{3}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 30 & 0 & 0\\5 & 0 & 0 & 0 & 0 & 75 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 30\\0 & 0 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 5 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 5 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 8 & & 71 & & 8 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P8 / O(2)+O(2)
- Ambient Space:
$A8$ - Dimension: \(6\)
- Ambient Index: \(5\)
- Basis Rank: \(7\)
- Eigenvalues: \(5 \cdot 2^{\frac{4}{5}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{4}{5}} \sqrt{5}}{4} - \frac{5 \cdot 2^{\frac{4}{5}}}{4} - 5 \cdot 2^{\frac{4}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{4}{5}} \sqrt{5}}{4} - \frac{5 \cdot 2^{\frac{4}{5}}}{4} + 5 \cdot 2^{\frac{4}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{4}{5}}}{4} + \frac{5 \cdot 2^{\frac{4}{5}} \sqrt{5}}{4} - 5 \cdot 2^{\frac{4}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{4}{5}}}{4} + \frac{5 \cdot 2^{\frac{4}{5}} \sqrt{5}}{4} + 5 \cdot 2^{\frac{4}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 20 & 0 & 0\\5 & 0 & 0 & 0 & 0 & 40 & 0\\0 & 5 & 0 & 0 & 0 & 0 & 20\\0 & 0 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 5 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 5 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 0 & & 10 & & 0 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P7 / O(4)
- Ambient Space:
$A7$ - Dimension: \(6\)
- Ambient Index: \(4\)
- Basis Rank: \(7\)
- Eigenvalues: \(-16\) (mult: 1), \(16\) (mult: 1), \(- 16 i\) (mult: 1), \(16 i\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 96 & 0 & 0 & 0\\4 & 0 & 0 & 0 & 416 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 416 & 0\\0 & 0 & 4 & 0 & 0 & 0 & 96\\0 & 0 & 0 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 1 & & 266 & & 1108 & & 266 & & 1 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P8 / O(2)+O(3)
- Ambient Space:
$A8$ - Dimension: \(6\)
- Ambient Index: \(4\)
- Basis Rank: \(7\)
- Eigenvalues: \(- 4 \sqrt{2} \cdot 3^{\frac{3}{4}}\) (mult: 1), \(4 \sqrt{2} \cdot 3^{\frac{3}{4}}\) (mult: 1), \(- 4 \sqrt{2} \cdot 3^{\frac{3}{4}} i\) (mult: 1), \(4 \sqrt{2} \cdot 3^{\frac{3}{4}} i\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 48 & 0 & 0 & 0\\4 & 0 & 0 & 0 & 168 & 0 & 0\\0 & 4 & 0 & 0 & 0 & 168 & 0\\0 & 0 & 4 & 0 & 0 & 0 & 48\\0 & 0 & 0 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 45 & & 252 & & 45 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P7 / O(5)
- Ambient Space:
$A7$ - Dimension: \(6\)
- Ambient Index: \(3\)
- Basis Rank: \(7\)
- Eigenvalues: \(15 \cdot 5^{\frac{2}{3}}\) (mult: 1), \(- \frac{15 \cdot 5^{\frac{2}{3}}}{2} - \frac{15 \sqrt{3} \cdot 5^{\frac{2}{3}} i}{2}\) (mult: 1), \(- \frac{15 \cdot 5^{\frac{2}{3}}}{2} + \frac{15 \sqrt{3} \cdot 5^{\frac{2}{3}} i}{2}\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 360 & 0 & 0 & 298800 & 0\\3 & 0 & 0 & 2310 & 0 & 0 & 298800\\0 & 3 & 0 & 0 & 4035 & 0 & 0\\0 & 0 & 3 & 0 & 0 & 2310 & 0\\0 & 0 & 0 & 3 & 0 & 0 & 360\\0 & 0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 36 & & 2472 & & 8093 & & 2472 & & 36 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P8 / O(2)+O(4)
- Ambient Space:
$A8$ - Dimension: \(6\)
- Ambient Index: \(3\)
- Basis Rank: \(7\)
- Eigenvalues: \(24 \sqrt[3]{2}\) (mult: 1), \(- 12 \sqrt[3]{2} - 12 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(- 12 \sqrt[3]{2} + 12 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 144 & 0 & 0 & 40320 & 0\\3 & 0 & 0 & 768 & 0 & 0 & 40320\\0 & 3 & 0 & 0 & 1248 & 0 & 0\\0 & 0 & 3 & 0 & 0 & 768 & 0\\0 & 0 & 0 & 3 & 0 & 0 & 144\\0 & 0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 9 & & 882 & & 3140 & & 882 & & 9 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P8 / O(3)+O(3)
- Ambient Space:
$A8$ - Dimension: \(6\)
- Ambient Index: \(3\)
- Basis Rank: \(7\)
- Eigenvalues: \(27\) (mult: 1), \(- \frac{27}{2} - \frac{27 \sqrt{3} i}{2}\) (mult: 1), \(- \frac{27}{2} + \frac{27 \sqrt{3} i}{2}\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 108 & 0 & 0 & 21384 & 0\\3 & 0 & 0 & 540 & 0 & 0 & 21384\\0 & 3 & 0 & 0 & 891 & 0 & 0\\0 & 0 & 3 & 0 & 0 & 540 & 0\\0 & 0 & 0 & 3 & 0 & 0 & 108\\0 & 0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 2 & & 329 & & 1303 & & 329 & & 2 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P7 / O(6)
- Ambient Space:
$A7$ - Dimension: \(6\)
- Ambient Index: \(2\)
- Basis Rank: \(7\)
- Eigenvalues: \(-432\) (mult: 1), \(432\) (mult: 1), \(0\) (mult: 5)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 1440 & 0 & 37687680 & 0 & 288139991040 & 0\\2 & 0 & 12528 & 0 & 262916928 & 0 & 288139991040\\0 & 2 & 0 & 32688 & 0 & 262916928 & 0\\0 & 0 & 2 & 0 & 32688 & 0 & 37687680\\0 & 0 & 0 & 2 & 0 & 12528 & 0\\0 & 0 & 0 & 0 & 2 & 0 & 1440\\0 & 0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 330 & & 13140 & & 38166 & & 13140 & & 330 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P8 / O(2)+O(5)
- Ambient Space:
$A8$ - Dimension: \(6\)
- Ambient Index: \(2\)
- Basis Rank: \(7\)
- Eigenvalues: \(- 100 \sqrt{5}\) (mult: 1), \(100 \sqrt{5}\) (mult: 1), \(0\) (mult: 5)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 480 & 0 & 3273600 & 0 & 7293600000 & 0\\2 & 0 & 3560 & 0 & 19926400 & 0 & 7293600000\\0 & 2 & 0 & 8460 & 0 & 19926400 & 0\\0 & 0 & 2 & 0 & 8460 & 0 & 3273600\\0 & 0 & 0 & 2 & 0 & 3560 & 0\\0 & 0 & 0 & 0 & 2 & 0 & 480\\0 & 0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 157 & & 6938 & & 20764 & & 6938 & & 157 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
P8 / O(3)+O(4)
- Ambient Space:
$A8$ - Dimension: \(6\)
- Ambient Index: \(2\)
- Basis Rank: \(7\)
- Eigenvalues: \(- 96 \sqrt{3}\) (mult: 1), \(96 \sqrt{3}\) (mult: 1), \(0\) (mult: 5)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 288 & 0 & 1081728 & 0 & 1344190464 & 0\\2 & 0 & 1968 & 0 & 6113088 & 0 & 1344190464\\0 & 2 & 0 & 4656 & 0 & 6113088 & 0\\0 & 0 & 2 & 0 & 4656 & 0 & 1081728\\0 & 0 & 0 & 2 & 0 & 1968 & 0\\0 & 0 & 0 & 0 & 2 & 0 & 288\\0 & 0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & & 1 & & & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
0 & & 54 & & 2720 & & 8534 & & 2720 & & 54 & & 0 \\
& 0 & & 0 & & 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 0 & & 1 & & 0 & & 0 & & \\
& & & 0 & & 0 & & 0 & & 0 & & & \\
& & & & 0 & & 1 & & 0 & & & & \\
& & & & & 0 & & 0 & & & & & \\
& & & & & & 1 & & & & & & \\
\end{matrix}
\]
5-folds¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P5 Fano index 6 | 6 | 6 |
| P6 / O(2) | 5 | 6 |
| P6 / O(3) | 4 | 6 |
| P7 / O(2)+O(2) | 4 | 6 |
| P6 / O(4) | 3 | 6 |
| P7 / O(2)+O(3) | 3 | 6 |
| P8 / O(2)+O(2)+O(2) | 3 | 6 |
| P6 / O(5) | 2 | 6 |
| P7 / O(2)+O(4) | 2 | 6 |
| P7 / O(3)+O(3) | 2 | 6 |
| P8 / O(2)+O(2)+O(3) | 2 | 6 |
P5 Fano index 6
- Ambient Space:
$A5$ - Dimension: \(5\)
- Ambient Index: \(6\)
- Basis Rank: \(6\)
- Eigenvalues: \(-6\) (mult: 1), \(6\) (mult: 1), \(-3 - 3 \sqrt{3} i\) (mult: 1), \(-3 + 3 \sqrt{3} i\) (mult: 1), \(3 - 3 \sqrt{3} i\) (mult: 1), \(3 + 3 \sqrt{3} i\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 6\\6 & 0 & 0 & 0 & 0 & 0\\0 & 6 & 0 & 0 & 0 & 0\\0 & 0 & 6 & 0 & 0 & 0\\0 & 0 & 0 & 6 & 0 & 0\\0 & 0 & 0 & 0 & 6 & 0\end{bmatrix}
\]
P6 / O(2)
- Ambient Space:
$A6$ - Dimension: \(5\)
- Ambient Index: \(5\)
- Basis Rank: \(6\)
- Eigenvalues: \(5 \cdot 2^{\frac{2}{5}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{5 \cdot 2^{\frac{2}{5}}}{4} - 5 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{5 \cdot 2^{\frac{2}{5}}}{4} + 5 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{2}{5}}}{4} + \frac{5 \cdot 2^{\frac{2}{5}} \sqrt{5}}{4} - 5 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(- \frac{5 \cdot 2^{\frac{2}{5}}}{4} + \frac{5 \cdot 2^{\frac{2}{5}} \sqrt{5}}{4} + 5 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(0\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 10 & 0\\5 & 0 & 0 & 0 & 0 & 10\\0 & 5 & 0 & 0 & 0 & 0\\0 & 0 & 5 & 0 & 0 & 0\\0 & 0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 0 & 5 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 0 & & 0 & & 0 & & 0 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P6 / O(3)
- Ambient Space:
$A6$ - Dimension: \(5\)
- Ambient Index: \(4\)
- Basis Rank: \(6\)
- Eigenvalues: \(- 4 \cdot 3^{\frac{3}{4}}\) (mult: 1), \(4 \cdot 3^{\frac{3}{4}}\) (mult: 1), \(- 4 \cdot 3^{\frac{3}{4}} i\) (mult: 1), \(4 \cdot 3^{\frac{3}{4}} i\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 24 & 0 & 0\\4 & 0 & 0 & 0 & 60 & 0\\0 & 4 & 0 & 0 & 0 & 24\\0 & 0 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 0 & & 21 & & 21 & & 0 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P7 / O(2)+O(2)
- Ambient Space:
$A7$ - Dimension: \(5\)
- Ambient Index: \(4\)
- Basis Rank: \(6\)
- Eigenvalues: \(-8\) (mult: 1), \(8\) (mult: 1), \(- 8 i\) (mult: 1), \(8 i\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 16 & 0 & 0\\4 & 0 & 0 & 0 & 32 & 0\\0 & 4 & 0 & 0 & 0 & 16\\0 & 0 & 4 & 0 & 0 & 0\\0 & 0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 0 & & 3 & & 3 & & 0 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P6 / O(4)
- Ambient Space:
$A6$ - Dimension: \(5\)
- Ambient Index: \(3\)
- Basis Rank: \(6\)
- Eigenvalues: \(12 \cdot 2^{\frac{2}{3}}\) (mult: 1), \(- 6 \cdot 2^{\frac{2}{3}} - 6 \cdot 2^{\frac{2}{3}} \sqrt{3} i\) (mult: 1), \(- 6 \cdot 2^{\frac{2}{3}} + 6 \cdot 2^{\frac{2}{3}} \sqrt{3} i\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 72 & 0 & 0 & 1728\\3 & 0 & 0 & 312 & 0 & 0\\0 & 3 & 0 & 0 & 312 & 0\\0 & 0 & 3 & 0 & 0 & 72\\0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 7 & & 266 & & 266 & & 7 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P7 / O(2)+O(3)
- Ambient Space:
$A7$ - Dimension: \(5\)
- Ambient Index: \(3\)
- Basis Rank: \(6\)
- Eigenvalues: \(9 \cdot 2^{\frac{2}{3}}\) (mult: 1), \(- \frac{9 \cdot 2^{\frac{2}{3}}}{2} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}\) (mult: 1), \(- \frac{9 \cdot 2^{\frac{2}{3}}}{2} + \frac{9 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 36 & 0 & 0 & 432\\3 & 0 & 0 & 126 & 0 & 0\\0 & 3 & 0 & 0 & 126 & 0\\0 & 0 & 3 & 0 & 0 & 36\\0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 1 & & 83 & & 83 & & 1 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P8 / O(2)+O(2)+O(2)
- Ambient Space:
$A8$ - Dimension: \(5\)
- Ambient Index: \(3\)
- Basis Rank: \(6\)
- Eigenvalues: \(12\) (mult: 1), \(-6 - 6 \sqrt{3} i\) (mult: 1), \(-6 + 6 \sqrt{3} i\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 24 & 0 & 0 & 192\\3 & 0 & 0 & 72 & 0 & 0\\0 & 3 & 0 & 0 & 72 & 0\\0 & 0 & 3 & 0 & 0 & 24\\0 & 0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 0 & & 27 & & 27 & & 0 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P6 / O(5)
- Ambient Space:
$A6$ - Dimension: \(5\)
- Ambient Index: \(2\)
- Basis Rank: \(6\)
- Eigenvalues: \(- 50 \sqrt{5}\) (mult: 1), \(50 \sqrt{5}\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 240 & 0 & 422400 & 0 & 62640000\\2 & 0 & 1540 & 0 & 1385000 & 0\\0 & 2 & 0 & 2690 & 0 & 422400\\0 & 0 & 2 & 0 & 1540 & 0\\0 & 0 & 0 & 2 & 0 & 240\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 84 & & 1554 & & 1554 & & 84 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P7 / O(2)+O(4)
- Ambient Space:
$A7$ - Dimension: \(5\)
- Ambient Index: \(2\)
- Basis Rank: \(6\)
- Eigenvalues: \(-64\) (mult: 1), \(64\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 96 & 0 & 53376 & 0 & 3207168\\2 & 0 & 512 & 0 & 157952 & 0\\0 & 2 & 0 & 832 & 0 & 53376\\0 & 0 & 2 & 0 & 512 & 0\\0 & 0 & 0 & 2 & 0 & 96\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 36 & & 783 & & 783 & & 36 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P7 / O(3)+O(3)
- Ambient Space:
$A7$ - Dimension: \(5\)
- Ambient Index: \(2\)
- Basis Rank: \(6\)
- Eigenvalues: \(-54\) (mult: 1), \(54\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 72 & 0 & 28512 & 0 & 1283040\\2 & 0 & 360 & 0 & 79056 & 0\\0 & 2 & 0 & 594 & 0 & 28512\\0 & 0 & 2 & 0 & 360 & 0\\0 & 0 & 0 & 2 & 0 & 72\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 16 & & 410 & & 410 & & 16 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
P8 / O(2)+O(2)+O(3)
- Ambient Space:
$A8$ - Dimension: \(5\)
- Ambient Index: \(2\)
- Basis Rank: \(6\)
- Eigenvalues: \(- 24 \sqrt{3}\) (mult: 1), \(24 \sqrt{3}\) (mult: 1), \(0\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 48 & 0 & 10944 & 0 & 331776\\2 & 0 & 216 & 0 & 29088 & 0\\0 & 2 & 0 & 336 & 0 & 10944\\0 & 0 & 2 & 0 & 216 & 0\\0 & 0 & 0 & 2 & 0 & 48\\0 & 0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & & 1 & & & & & \\
& & & & 0 & & 0 & & & & \\
& & & 0 & & 1 & & 0 & & & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
0 & & 11 & & 316 & & 316 & & 11 & & 0 \\
& 0 & & 0 & & 1 & & 0 & & 0 & \\
& & 0 & & 0 & & 0 & & 0 & & \\
& & & 0 & & 1 & & 0 & & & \\
& & & & 0 & & 0 & & & & \\
& & & & & 1 & & & & & \\
\end{matrix}
\]
4-folds¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P4 Fano index 5 | 5 | 5 |
| P5 / O(2) | 4 | 5 |
| P5 / O(3) | 3 | 5 |
| P6 / O(2)+O(2) | 3 | 5 |
| P5 / O(4) | 2 | 5 |
| P6 / O(2)+O(3) | 2 | 5 |
| P7 / O(2)+O(2)+O(2) | 2 | 5 |
| P5 / O(5) | 1 | 5 |
| P6 / O(3)+O(3) | 1 | 5 |
| P6 / O(2)+O(4) | 1 | 5 |
| P7 / O(2)+O(2)+O(3) | 1 | 5 |
P4 Fano index 5
- Ambient Space:
$A4$ - Dimension: \(4\)
- Ambient Index: \(5\)
- Basis Rank: \(5\)
- Eigenvalues: \(5\) (mult: 1), \(- \frac{5 \sqrt{5}}{4} - \frac{5}{4} - 5 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5 \sqrt{5}}{4} - \frac{5}{4} + 5 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\) (mult: 1), \(- \frac{5}{4} + \frac{5 \sqrt{5}}{4} - 5 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1), \(- \frac{5}{4} + \frac{5 \sqrt{5}}{4} + 5 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 0 & 5\\5 & 0 & 0 & 0 & 0\\0 & 5 & 0 & 0 & 0\\0 & 0 & 5 & 0 & 0\\0 & 0 & 0 & 5 & 0\end{bmatrix}
\]
P5 / O(2)
- Ambient Space:
$A5$ - Dimension: \(4\)
- Ambient Index: \(4\)
- Basis Rank: \(5\)
- Eigenvalues: \(- 4 \sqrt{2}\) (mult: 1), \(4 \sqrt{2}\) (mult: 1), \(- 4 \sqrt{2} i\) (mult: 1), \(4 \sqrt{2} i\) (mult: 1), \(0\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 8 & 0\\4 & 0 & 0 & 0 & 8\\0 & 4 & 0 & 0 & 0\\0 & 0 & 4 & 0 & 0\\0 & 0 & 0 & 4 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 2 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P5 / O(3)
- Ambient Space:
$A5$ - Dimension: \(4\)
- Ambient Index: \(3\)
- Basis Rank: \(5\)
- Eigenvalues: \(9\) (mult: 1), \(- \frac{9}{2} - \frac{9 \sqrt{3} i}{2}\) (mult: 1), \(- \frac{9}{2} + \frac{9 \sqrt{3} i}{2}\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 18 & 0 & 0\\3 & 0 & 0 & 45 & 0\\0 & 3 & 0 & 0 & 18\\0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 1 & & 21 & & 1 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P6 / O(2)+O(2)
- Ambient Space:
$A6$ - Dimension: \(4\)
- Ambient Index: \(3\)
- Basis Rank: \(5\)
- Eigenvalues: \(6 \sqrt[3]{2}\) (mult: 1), \(- 3 \sqrt[3]{2} - 3 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(- 3 \sqrt[3]{2} + 3 \sqrt[3]{2} \sqrt{3} i\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 12 & 0 & 0\\3 & 0 & 0 & 24 & 0\\0 & 3 & 0 & 0 & 12\\0 & 0 & 3 & 0 & 0\\0 & 0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 0 & & 8 & & 0 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P5 / O(4)
- Ambient Space:
$A5$ - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(5\)
- Eigenvalues: \(-32\) (mult: 1), \(32\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 48 & 0 & 5568 & 0\\2 & 0 & 208 & 0 & 5568\\0 & 2 & 0 & 208 & 0\\0 & 0 & 2 & 0 & 48\\0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 21 & & 142 & & 21 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P6 / O(2)+O(3)
- Ambient Space:
$A6$ - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(5\)
- Eigenvalues: \(- 12 \sqrt{3}\) (mult: 1), \(12 \sqrt{3}\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 24 & 0 & 1152 & 0\\2 & 0 & 84 & 0 & 1152\\0 & 2 & 0 & 84 & 0\\0 & 0 & 2 & 0 & 24\\0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 8 & & 70 & & 8 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P7 / O(2)+O(2)+O(2)
- Ambient Space:
$A7$ - Dimension: \(4\)
- Ambient Index: \(2\)
- Basis Rank: \(5\)
- Eigenvalues: \(-16\) (mult: 1), \(16\) (mult: 1), \(0\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 16 & 0 & 448 & 0\\2 & 0 & 48 & 0 & 448\\0 & 2 & 0 & 48 & 0\\0 & 0 & 2 & 0 & 16\\0 & 0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 3 & & 38 & & 3 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P5 / O(5)
- Ambient Space:
$A5$ - Dimension: \(4\)
- Ambient Index: \(1\)
- Basis Rank: \(5\)
- Eigenvalues: \(3005\) (mult: 1), \(-120\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 198000 & 395550000 & 423858600000 & 109236016800000\\1 & 650 & 1487500 & 1620307500 & 423858600000\\0 & 1 & 1225 & 1487500 & 395550000\\0 & 0 & 1 & 650 & 198000\\0 & 0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 120 & & 581 & & 120 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P6 / O(3)+O(3)
- Ambient Space:
$A6$ - Dimension: \(4\)
- Ambient Index: \(1\)
- Basis Rank: \(5\)
- Eigenvalues: \(693\) (mult: 1), \(-36\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 13608 & 6244776 & 1637648928 & 122790427200\\1 & 144 & 80352 & 21464352 & 1637648928\\0 & 1 & 261 & 80352 & 6244776\\0 & 0 & 1 & 144 & 13608\\0 & 0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 49 & & 267 & & 49 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P6 / O(2)+O(4)
- Ambient Space:
$A6$ - Dimension: \(4\)
- Ambient Index: \(1\)
- Basis Rank: \(5\)
- Eigenvalues: \(976\) (mult: 1), \(-48\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 25632 & 16464384 & 6088232448 & 607435407360\\1 & 208 & 159328 & 60008448 & 6088232448\\0 & 1 & 368 & 159328 & 16464384\\0 & 0 & 1 & 208 & 25632\\0 & 0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 77 & & 394 & & 77 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
P7 / O(2)+O(2)+O(3)
- Ambient Space:
$A7$ - Dimension: \(4\)
- Ambient Index: \(1\)
- Basis Rank: \(5\)
- Eigenvalues: \(408\) (mult: 1), \(-24\) (mult: 4)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 5328 & 1427328 & 230750208 & 11257159680\\1 & 84 & 28152 & 4643136 & 230750208\\0 & 1 & 144 & 28152 & 1427328\\0 & 0 & 1 & 84 & 5328\\0 & 0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & & 1 & & & & \\
& & & 0 & & 0 & & & \\
& & 0 & & 1 & & 0 & & \\
& 0 & & 0 & & 0 & & 0 & \\
0 & & 42 & & 236 & & 42 & & 0 \\
& 0 & & 0 & & 0 & & 0 & \\
& & 0 & & 1 & & 0 & & \\
& & & 0 & & 0 & & & \\
& & & & 1 & & & & \\
\end{matrix}
\]
3-folds¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P3 Fano index 4 | 4 | 4 |
| P4 / O(2) | 3 | 4 |
| P4 / O(3) | 2 | 4 |
| P5 / O(2)+O(2) | 2 | 4 |
| P4 / O(4) | 1 | 4 |
| P5 / O(2)+O(3) | 1 | 4 |
| P6 / O(2)+O(2)+O(2) | 1 | 4 |
P3 Fano index 4
- Ambient Space:
$A3$ - Dimension: \(3\)
- Ambient Index: \(4\)
- Basis Rank: \(4\)
- Eigenvalues: \(-4\) (mult: 1), \(4\) (mult: 1), \(- 4 i\) (mult: 1), \(4 i\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 0 & 4\\4 & 0 & 0 & 0\\0 & 4 & 0 & 0\\0 & 0 & 4 & 0\end{bmatrix}
\]
P4 / O(2)
- Ambient Space:
$A4$ - Dimension: \(3\)
- Ambient Index: \(3\)
- Basis Rank: \(4\)
- Eigenvalues: \(3 \cdot 2^{\frac{2}{3}}\) (mult: 1), \(- \frac{3 \cdot 2^{\frac{2}{3}}}{2} - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}\) (mult: 1), \(- \frac{3 \cdot 2^{\frac{2}{3}}}{2} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{3} i}{2}\) (mult: 1), \(0\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 6 & 0\\3 & 0 & 0 & 6\\0 & 3 & 0 & 0\\0 & 0 & 3 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & 1 & & & \\
& & 0 & & 0 & & \\
& 0 & & 1 & & 0 & \\
0 & & 0 & & 0 & & 0 \\
& 0 & & 1 & & 0 & \\
& & 0 & & 0 & & \\
& & & 1 & & & \\
\end{matrix}
\]
P4 / O(3)
- Ambient Space:
$A4$ - Dimension: \(3\)
- Ambient Index: \(2\)
- Basis Rank: \(4\)
- Eigenvalues: \(- 6 \sqrt{3}\) (mult: 1), \(6 \sqrt{3}\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 12 & 0 & 72\\2 & 0 & 30 & 0\\0 & 2 & 0 & 12\\0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & 1 & & & \\
& & 0 & & 0 & & \\
& 0 & & 1 & & 0 & \\
0 & & 5 & & 5 & & 0 \\
& 0 & & 1 & & 0 & \\
& & 0 & & 0 & & \\
& & & 1 & & & \\
\end{matrix}
\]
P5 / O(2)+O(2)
- Ambient Space:
$A5$ - Dimension: \(3\)
- Ambient Index: \(2\)
- Basis Rank: \(4\)
- Eigenvalues: \(-8\) (mult: 1), \(8\) (mult: 1), \(0\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 8 & 0 & 32\\2 & 0 & 16 & 0\\0 & 2 & 0 & 8\\0 & 0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & 1 & & & \\
& & 0 & & 0 & & \\
& 0 & & 1 & & 0 & \\
0 & & 2 & & 2 & & 0 \\
& 0 & & 1 & & 0 & \\
& & 0 & & 0 & & \\
& & & 1 & & & \\
\end{matrix}
\]
P4 / O(4)
- Ambient Space:
$A4$ - Dimension: \(3\)
- Ambient Index: \(1\)
- Basis Rank: \(4\)
- Eigenvalues: \(232\) (mult: 1), \(-24\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 3888 & 504576 & 18323712\\1 & 80 & 13600 & 504576\\0 & 1 & 80 & 3888\\0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & 1 & & & \\
& & 0 & & 0 & & \\
& 0 & & 1 & & 0 & \\
0 & & 30 & & 30 & & 0 \\
& 0 & & 1 & & 0 & \\
& & 0 & & 0 & & \\
& & & 1 & & & \\
\end{matrix}
\]
P5 / O(2)+O(3)
- Ambient Space:
$A5$ - Dimension: \(3\)
- Ambient Index: \(1\)
- Basis Rank: \(4\)
- Eigenvalues: \(96\) (mult: 1), \(-12\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 792 & 43632 & 793152\\1 & 30 & 2340 & 43632\\0 & 1 & 30 & 792\\0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & 1 & & & \\
& & 0 & & 0 & & \\
& 0 & & 1 & & 0 & \\
0 & & 20 & & 20 & & 0 \\
& 0 & & 1 & & 0 & \\
& & 0 & & 0 & & \\
& & & 1 & & & \\
\end{matrix}
\]
P6 / O(2)+O(2)+O(2)
- Ambient Space:
$A6$ - Dimension: \(3\)
- Ambient Index: \(1\)
- Basis Rank: \(4\)
- Eigenvalues: \(56\) (mult: 1), \(-8\) (mult: 3)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 304 & 9984 & 121088\\1 & 16 & 800 & 9984\\0 & 1 & 16 & 304\\0 & 0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & & 1 & & & \\
& & 0 & & 0 & & \\
& 0 & & 1 & & 0 & \\
0 & & 14 & & 14 & & 0 \\
& 0 & & 1 & & 0 & \\
& & 0 & & 0 & & \\
& & & 1 & & & \\
\end{matrix}
\]
Surfaces (2-folds)¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P2 Fano index 3 | 3 | 3 |
| P3 / O(2) | 2 | 3 |
| P3 / O(3) | 1 | 3 |
| P4 / O(2)+O(2) | 1 | 3 |
P2 Fano index 3
- Ambient Space:
$A2$ - Dimension: \(2\)
- Ambient Index: \(3\)
- Basis Rank: \(3\)
- Eigenvalues: \(3\) (mult: 1), \(- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}\) (mult: 1), \(- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 0 & 3\\3 & 0 & 0\\0 & 3 & 0\end{bmatrix}
\]
P3 / O(2)
- Ambient Space:
$A3$ - Dimension: \(2\)
- Ambient Index: \(2\)
- Basis Rank: \(3\)
- Eigenvalues: \(-4\) (mult: 1), \(4\) (mult: 1), \(0\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 4 & 0\\2 & 0 & 4\\0 & 2 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & 1 & & \\
& 0 & & 0 & \\
0 & & 2 & & 0 \\
& 0 & & 0 & \\
& & 1 & & \\
\end{matrix}
\]
P3 / O(3)
- Ambient Space:
$A3$ - Dimension: \(2\)
- Ambient Index: \(1\)
- Basis Rank: \(3\)
- Eigenvalues: \(21\) (mult: 1), \(-6\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 108 & 756\\1 & 9 & 108\\0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & 1 & & \\
& 0 & & 0 & \\
0 & & 7 & & 0 \\
& 0 & & 0 & \\
& & 1 & & \\
\end{matrix}
\]
P4 / O(2)+O(2)
- Ambient Space:
$A4$ - Dimension: \(2\)
- Ambient Index: \(1\)
- Basis Rank: \(3\)
- Eigenvalues: \(12\) (mult: 1), \(-4\) (mult: 2)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 40 & 192\\1 & 4 & 40\\0 & 1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& & 1 & & \\
& 0 & & 0 & \\
0 & & 6 & & 0 \\
& 0 & & 0 & \\
& & 1 & & \\
\end{matrix}
\]
Curves (1-folds)¶
| Geometry | Ambient Index | Basis Rank |
|---|---|---|
| P2 / O(2) | 1 | 2 |
P2 / O(2)
- Ambient Space:
$A2$ - Dimension: \(1\)
- Ambient Index: \(1\)
- Basis Rank: \(2\)
- Eigenvalues: \(-2\) (mult: 1), \(2\) (mult: 1)
Quantum Matrix (\(y=1\))
\[
\begin{bmatrix}0 & 4\\1 & 0\end{bmatrix}
\]
Hodge Diamond (\(h^{p,q}\))
\[
\begin{matrix}
& 1 & \\
0 & & 0 \\
& 1 & \\
\end{matrix}
\]