Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-195-45-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ8P8
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_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11node 12: fundamental weight ω_12ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12
Bundle
2 copies of E^(w4 + w5) + 2 copies of E^(w2 + w5)
Bundle
2 copies of Q_2 box O_3(1) + 2 copies of Q_1 box O_3(1)
ZL62 label
2020800.w0Uw0Uw0Dw0D
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;9) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;8) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 195
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 195
$\mathrm{c}_1^2\mathrm{c}_2$ = 138
$\mathrm{c}_2^2$ = 111
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 10 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 32 0 0 x1 - 4 36
$\mathrm{HH}^{ 4 }$ = 45 0 0 0 0 45
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 113t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(220)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(298)

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( t^{2} + 11t - 1 \right) \left( 27t^{2} + 1 \right) \left( 73t^{4} + 49t^{3} + 56t^{2} + 14t + 1 \right) \left( 611156365000t^{16} + 3822794295950t^{15} + 1332583918100t^{14} - 3619526087540t^{13} - 12477568080400t^{12} - 29745773508555t^{11} - 27821407883534t^{10} - 16711320145929t^{9} - 7448125033211t^{8} - 2605348044940t^{7} - 572413087530t^{6} - 59847435634t^{5} - 5211129618t^{4} - 633258127t^{3} - 18043838t^{2} - 160373t - 3445 \right)D^{6} + \left( 16864248735810000t^{24} + 253611323564361750t^{23} + 1061309039603303400t^{22} + 645896182585001570t^{21} - 1188305649535860530t^{20} - 5519814447747527865t^{19} - 13437807303712961880t^{18} - 16822129076370640450t^{17} - 15106646056203669255t^{16} - 10231713403481947444t^{15} - 5465947988266148897t^{14} - 2308483162036270815t^{13} - 777018245315726867t^{12} - 193851765222009376t^{11} - 32683699722550707t^{10} - 4383088205576161t^{9} - 591982469097175t^{8} - 77339955693186t^{7} - 8822703150087t^{6} - 666844610523t^{5} - 39238950783t^{4} - 3241249915t^{3} - 73182473t^{2} - 460449t - 6890 \right)D^{5} + \left( 95028703193850000t^{24} + 1233659440814103750t^{23} + 4519885789821402800t^{22} + 1946333568355028470t^{21} - 7567422890037327510t^{20} - 30543658420186825795t^{19} - 73970137585181895966t^{18} - 89337661231653742526t^{17} - 74459926768017979018t^{16} - 47534980881622077318t^{15} - 24535045904953911622t^{14} - 9961577143209217596t^{13} - 3165932927784273696t^{12} - 743258158993820208t^{11} - 110673112042514011t^{10} - 12218648003640170t^{9} - 1312923957666401t^{8} - 75955437922880t^{7} + 1320022456970t^{6} + 296245864122t^{5} + 16328527128t^{4} + 2857925335t^{3} + 64447177t^{2} + 429184t + 3445 \right)D^{4} + t \left( 275717082506100000t^{23} + 3179694939583199250t^{22} + 9899482525501623400t^{21} + 2655001244324734270t^{20} - 21227689187598409350t^{19} - 85779163021006413495t^{18} - 210470976901783947488t^{17} - 250627756326436618574t^{16} - 201950727941766048455t^{15} - 124336921830245191052t^{14} - 62253602315277927169t^{13} - 24017666378867240587t^{12} - 7041515397455602711t^{11} - 1586341271754634732t^{10} - 231831604937016425t^{9} - 27048688688727001t^{8} - 3530478371139313t^{7} - 268636329335816t^{6} - 11606680689559t^{5} - 277076605349t^{4} - 9857907675t^{3} - 253737575t^{2} - 4198391t - 1643 \right)D^{3} + 2 t ^{2} \left( 216892976796667500t^{22} + 2280989198014073400t^{21} + 5914049403129648450t^{20} + 522016329437725530t^{19} - 15869113250782577460t^{18} - 65364562491688731685t^{17} - 162394945816084807534t^{16} - 192486457712870335429t^{15} - 154332420995660656193t^{14} - 93532429983672392525t^{13} - 45754784412787852946t^{12} - 16765827096483393516t^{11} - 4434814883498129395t^{10} - 950892044090502880t^{9} - 138300177911857225t^{8} - 16251979843524883t^{7} - 2251904697915681t^{6} - 151212943061019t^{5} - 4924540995874t^{4} - 102434114678t^{3} - 5039914075t^{2} - 137232507t - 1628575 \right)D^{2} + 4 t ^{2} \left( 87466559911522500t^{22} + 857854653279591750t^{21} + 1834189617946922300t^{20} - 222368546671247550t^{19} - 6206581773062784790t^{18} - 25698398589348708185t^{17} - 64253719657169854180t^{16} - 76022247867851020815t^{15} - 61561907818529484435t^{14} - 37227805306760258795t^{13} - 17951418803842515194t^{12} - 6314491245702345662t^{11} - 1501325325322331999t^{10} - 304876746751239374t^{9} - 44913079717671772t^{8} - 5232582089582709t^{7} - 754203993031669t^{6} - 46091047113877t^{5} - 1240323368130t^{4} - 29489389388t^{3} - 1460174119t^{2} - 36672051t - 330720 \right)D + 8 t ^{2} \left( 14053540613175000t^{22} + 130936658855737500t^{21} + 231292638212794050t^{20} - 83341137931327950t^{19} - 989301391709842450t^{18} - 4056158407290142975t^{17} - 10147118001489479260t^{16} - 11986817451928973285t^{15} - 9855278629689849741t^{14} - 5986651779826950493t^{13} - 2865307542197693344t^{12} - 980234290955522318t^{11} - 211872234811050307t^{10} - 40685199001152118t^{9} - 6117876713541971t^{8} - 693399676207085t^{7} - 102409378897976t^{6} - 5862329966035t^{5} - 146131218243t^{4} - 4473996704t^{3} - 189739286t^{2} - 4033777t - 27560 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ8P8

|λ| = 11.09017 λ ≈ 11.09017; multiplicity 1; Jordan blocks 1 λ ≈ -6.699066 - 1.218204i; multiplicity 1; Jordan blocks 1 λ ≈ -6.699066 + 1.218204i; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 5.196152i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0 + 5.196152i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.3009344 - 1.218204i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3009344 + 1.218204i; multiplicity 1; Jordan blocks 1 λ ≈ -0.09016994; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 4; Jordan blocks 1, 1, 1, 1 2 2 4
Sector dimension
14
Spectral radius
$11.0901699$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.09017$11
$-6.699066 - 1.218204\mathrm{i}$11
$-6.699066 + 1.218204\mathrm{i}$11
$0 - 5.196152\mathrm{i}$21, 1
$0 + 5.196152\mathrm{i}$21, 1
$-0.3009344 - 1.218204\mathrm{i}$11
$-0.3009344 + 1.218204\mathrm{i}$11
$-0.09016994$11
$0$41, 1, 1, 1

Characteristic polynomial

$$\begin{equation}\lambda^{4} \left(\lambda^{2} + 27\right)^{2} \left(\lambda^{2} - 11 \lambda - 1\right) \left(\lambda^{4} + 14 \lambda^{3} + 56 \lambda^{2} + 49 \lambda + 73\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{2} + 27\right) \left(\lambda^{2} - 11 \lambda - 1\right) \left(\lambda^{4} + 14 \lambda^{3} + 56 \lambda^{2} + 49 \lambda + 73\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 4 & 3 & 9 & 6 & 9 & 6 & 21 & 48 & 24 & 24 & 80 \\ 1 & 0 & 0 & 2 & 2 & -2 & -2 & 4 & 4 & -2 & 0 & -3 & -3 & 0 \\ 1 & 0 & 0 & 2 & 2 & 4 & 4 & -2 & -2 & -2 & 0 & -3 & -3 & 0 \\ 1 & 0 & 0 & -1 & 0 & 4 & 2 & 4 & 2 & 12 & 12 & 9 & 9 & 48 \\ 0 & 1 & 1 & 0 & 0 & 2 & 0 & 2 & 0 & 4 & -8 & -2 & -2 & 12 \\ 0 & 1 & 0 & 1 & 0 & -1 & -2 & 0 & 0 & 0 & 4 & 2 & 0 & -12 \\ 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 3 & 4 & 4 & 2 & 12 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & -2 & 0 & 4 & 0 & 2 & -12 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 3 & 4 & 2 & 4 & 12 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & -1 & 0 & 0 & 0 & 12 \\ 0 & 0 & 0 & 0 & 1 & \frac{13}{4} & \frac{1}{4} & \frac{13}{4} & \frac{1}{4} & \frac{31}{4} & -1 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & -\frac{3}{2} & -\frac{1}{2} & \frac{3}{2} & \frac{5}{2} & -\frac{5}{2} & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{5}{2} & -\frac{3}{2} & -\frac{1}{2} & -\frac{5}{2} & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{4} & \frac{3}{4} & 0\end{pmatrix}\notag\end{equation}$$