Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-132-34-a

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
ℙ2P2 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
2 copies of E^(2w3) + E^(w2 + w3)
Bundle
2 copies of O_2(2) + Q_1 box O_2(1)
ZL62 label
20600.AAw7
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(1;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 34
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 132
$\chi(X, \mathrm{T}_X)$ = -16

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 132
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 2 0
0 1 1 0
0 0 10 0 0
0 1 1 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 16 0 x0 + 16 x2
$\mathrm{HH}^{ 3 }$ = x1 + 20 0 0 x1 20
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 15 0 0 x1 - x2 - 25 6 34
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 29t + 72t^{2} + 29t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(609)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(627)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 4t + 1 \right) ^{2} \left( 20t^{2} + 8t + 1 \right) ^{2} \left( 2160t^{3} - 432t^{2} + 36t - 1 \right) \left( 65210776704000t^{11} + 67363945286400t^{10} + 44699355244800t^{9} + 19821493146112t^{8} + 4586994862848t^{7} + 538787597216t^{6} + 32687215584t^{5} + 1485263600t^{4} + 40187804t^{3} + 2093459t^{2} - 91555t + 132 \right)D^{6} + 3 \left( 4t + 1 \right) \left( 20t^{2} + 8t + 1 \right) \left( 78878955501158400000t^{17} + 105834558489845760000t^{16} + 81797657266262016000t^{15} + 43472360378857881600t^{14} + 14244781802969026560t^{13} + 2673353671348856832t^{12} + 266344761214787584t^{11} + 11582082003125248t^{10} - 144373408046848t^{9} - 71515421933056t^{8} - 4300440293632t^{7} + 493256756352t^{6} + 33027020240t^{5} + 3247503976t^{4} + 12760856t^{3} + 3314068t^{2} - 91203t + 88 \right)D^{5} + \left( 157757911002316800000000t^{20} + 296380719604629504000000t^{19} + 296499097277060014080000t^{18} + 202894498224332679168000t^{17} + 97224267291038602444800t^{16} + 32259599236383212052480t^{15} + 7370677118992788819968t^{14} + 1155098226576426540032t^{13} + 123146093589110801408t^{12} + 8764887479124378624t^{11} + 384360583651521536t^{10} + 3360243134209984t^{9} - 1340932946433856t^{8} - 173779445404064t^{7} - 12886280391856t^{6} - 476336958108t^{5} - 12222479444t^{4} - 271325556t^{3} - 6956197t^{2} + 166326t - 132 \right)D^{4} + 4 t \left( 165645806552432640000000t^{19} + 296489147975442432000000t^{18} + 289298362436312371200000t^{17} + 194585101014242004480000t^{16} + 90462826806755751936000t^{15} + 28702759808169381273600t^{14} + 6231716817883858022400t^{13} + 933911966270022135552t^{12} + 97817822749377801728t^{11} + 7305184113980637952t^{10} + 395254833827054208t^{9} + 14646879834327696t^{8} + 262681177601856t^{7} - 10665918371728t^{6} - 1226571735504t^{5} - 5697287571t^{4} + 405008386t^{3} - 44713792t^{2} + 469093t + 104 \right)D^{3} + 4 t ^{2} \left( 365998353525374976000000t^{18} + 630231899806008115200000t^{17} + 603662621632423004160000t^{16} + 400424814541052637696000t^{15} + 181106977970691824025600t^{14} + 55192181615089716480000t^{13} + 11449181537947268455424t^{12} + 1647359536067214222848t^{11} + 168840564698502174976t^{10} + 12785817502614255360t^{9} + 733654287113425344t^{8} + 30690603945364560t^{7} + 749176841536320t^{6} - 12806235785160t^{5} - 1719857472972t^{4} - 6275046448t^{3} + 501771623t^{2} - 19117820t + 250206 \right)D^{2} + 16 t ^{2} \left( 99387483931459584000000t^{18} + 166191869095955251200000t^{17} + 157083855058931788800000t^{16} + 103085390040162399360000t^{15} + 45563046271596180864000t^{14} + 13427871477680004956160t^{13} + 2682884343791641156608t^{12} + 373365314428136192640t^{11} + 37562449640449578496t^{10} + 2856254356670822144t^{9} + 167692534251827232t^{8} + 7297520475644692t^{7} + 188030247618244t^{6} - 2838193452708t^{5} - 334430741956t^{4} - 1393120075t^{3} + 68881383t^{2} - 221680t + 17160 \right)D + 16 t ^{2} \left( 40566319972024320000000t^{18} + 66468795585601536000000t^{17} + 62268285936148254720000t^{16} + 40554140442759850752000t^{15} + 17619623540917290547200t^{14} + 5064339426060222904320t^{13} + 983819172800081267712t^{12} + 133545854306968574208t^{11} + 13247029186620125952t^{10} + 1007282102833433856t^{9} + 59589087064526464t^{8} + 2626680101701176t^{7} + 68574957274376t^{6} - 936308278176t^{5} - 98255277100t^{4} - 245017000t^{3} + 15357719t^{2} + 356600t + 2640 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ6P6

|λ| = 19.55953 λ ≈ 19.55953; multiplicity 1; Jordan blocks 1 λ ≈ 8.220237 - 6.546742i; multiplicity 1; Jordan blocks 1 λ ≈ 8.220237 + 6.546742i; multiplicity 1; Jordan blocks 1 λ ≈ -4 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4 + 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 2; Jordan blocks 2 2 2 2
Sector dimension
9
Spectral radius
$19.5595263$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.55953$11
$8.220237 - 6.546742\mathrm{i}$11
$8.220237 + 6.546742\mathrm{i}$11
$-4 - 2\mathrm{i}$21, 1
$-4 + 2\mathrm{i}$21, 1
$-4$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda^{2} + 8 \lambda + 20\right)^{2} \left(\lambda^{3} - 36 \lambda^{2} + 432 \lambda - 2160\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda^{2} + 8 \lambda + 20\right) \left(\lambda^{3} - 36 \lambda^{2} + 432 \lambda - 2160\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 40 & 12 & 12 & 204 & 192 & 160 & 960 \\ 2 & 0 & 4 & 40 & -2 & 56 & 192 & 0 & 32 \\ 1 & 0 & 4 & 0 & 2 & 40 & 24 & 24 & 160 \\ 0 & 1 & 2 & 4 & 0 & 8 & 40 & -2 & 0 \\ 0 & 2 & 0 & 4 & 0 & 4 & 56 & 40 & 192 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 12 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 8 & 4 & 40 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0\end{pmatrix}\notag\end{equation}$$