Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 5-168-40-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3 + w4 + w5) + E^(w1 + w2 + w5)
Bundle
O_3(1) box O_4(1) box O_5(1) + O_1(1) box O_2(1) box O_5(1)
ZL62 label
111120.x4G00x4200
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;2;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 40
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 168
$\chi(X, \mathrm{T}_X)$ = -2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 168
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 5 0
0 0 0 0
0 0 12 0 0
0 0 0 0
0 5 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 2 0 x0 + 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 19 0 0 x1 + 3 16
$\mathrm{HH}^{ 4 }$ = 40 0 0 0 0 40
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 35t + 96t^{2} + 35t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(507)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(555)

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) \left( 4t + 1 \right) ^{2} \left( 5t + 1 \right) ^{2} \left( 256t^{3} + 32t^{2} + 11t - 1 \right) \left( 7440593305600t^{11} + 14440601779200t^{10} + 10253820096256t^{9} + 3566341016576t^{8} + 698300763952t^{7} + 102856177040t^{6} + 17440267788t^{5} + 2637864032t^{4} + 222257205t^{3} + 7164027t^{2} - 90000t + 1260 \right)D^{6} + \left( 4t + 1 \right) \left( 5t + 1 \right) \left( 2590516965277696000t^{17} + 5769104741262950400t^{16} + 4937680337582161920t^{15} + 2178513673197879296t^{14} + 549392617759920128t^{13} + 85450454652590080t^{12} + 10439385605416960t^{11} + 1425997834026368t^{10} + 70556716410448t^{9} - 53193423882624t^{8} - 16285302696484t^{7} - 2194987432784t^{6} - 180520344529t^{5} - 12980992206t^{4} - 843219909t^{3} - 31792788t^{2} + 290160t - 2520 \right)D^{5} + \left( 350481707066982400000t^{19} + 928200862060249088000t^{18} + 1002301217890304000000t^{17} + 596090290349061898240t^{16} + 221238286280833974272t^{15} + 56125196003305590784t^{14} + 11034799319899440128t^{13} + 1976433018901192704t^{12} + 328556682236239552t^{11} + 40999465245531888t^{10} + 2476750458824272t^{9} - 152182641493612t^{8} - 43438019032824t^{7} - 3197661838979t^{6} + 14976098820t^{5} + 19719515990t^{4} + 1340300538t^{3} + 32534691t^{2} - 108720t + 1260 \right)D^{4} + t \left( 1203828472099635200000t^{18} + 3163604381964697600000t^{17} + 3353866447716483072000t^{16} + 1945998062591120179200t^{15} + 706247907073965342720t^{14} + 179182304572079620096t^{13} + 37011702859726010368t^{12} + 7148302331058392576t^{11} + 1226309395579974976t^{10} + 153149310900700400t^{9} + 10502845145068384t^{8} - 56171028615020t^{7} - 78649210269176t^{6} - 6581301798847t^{5} - 230159171589t^{4} - 2742782949t^{3} - 76060047t^{2} - 5178810t - 1980 \right)D^{3} + 2 t ^{2} \left( 1103255460506501120000t^{17} + 2883995737847758848000t^{16} + 3015293588308197376000t^{15} + 1715526288330709729280t^{14} + 611221498811271733248t^{13} + 155048451943737491456t^{12} + 33223008747034140160t^{11} + 6736937813334590592t^{10} + 1178662391553388224t^{9} + 148932865613347280t^{8} + 11389921529273120t^{7} + 316239562959280t^{6} - 22948357793432t^{5} - 2286614055053t^{4} - 71763939187t^{3} - 1779044729t^{2} - 94777314t - 1853250 \right)D^{2} + 8 t ^{2} \left( 254480196000808960000t^{17} + 662949586785796096000t^{16} + 686261068025724928000t^{15} + 384780556161203200000t^{14} + 135297799049922732032t^{13} + 34440882433707074560t^{12} + 7627997999971394176t^{11} + 1603784131001390432t^{10} + 283271985162807584t^{9} + 35672148245473568t^{8} + 2798517467228008t^{7} + 104242203981296t^{6} - 1205296877744t^{5} - 234629186844t^{4} - 6905224809t^{3} - 313547622t^{2} - 16527420t - 178920 \right)D + 96 t ^{2} \left( 7619167544934400000t^{17} + 19806548683128832000t^{16} + 20363894821691392000t^{15} + 11301665156152463360t^{14} + 3939932361130334208t^{13} + 1008652869884848640t^{12} + 229757783064227392t^{11} + 49598045194330224t^{10} + 8801253019761968t^{9} + 1100212626681516t^{8} + 86311000565132t^{7} + 3452731620658t^{6} + 6690470846t^{5} - 4187921514t^{4} - 144865629t^{3} - 8916366t^{2} - 364860t - 2520 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ2P2

|λ| = 14.44287 λ ≈ 14.44287; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 1, 1, 2 λ ≈ -1.721437 - 3.84209i; multiplicity 1; Jordan blocks 1 λ ≈ -1.721437 + 3.84209i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 8; Jordan blocks 1, 1, 1, 1, 1, 1, 1, 1 λ ≈ 4; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 4 8 4 2
Sector dimension
21
Spectral radius
$14.4428745$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14.44287$11
$-5$41, 1, 2
$-1.721437 - 3.84209\mathrm{i}$11
$-1.721437 + 3.84209\mathrm{i}$11
$-4$81, 1, 1, 1, 1, 1, 1, 1
$4$41, 1, 1, 1
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right)^{8} \left(\lambda + 5\right)^{4} \left(\lambda - 4\right)^{4} \left(\lambda^{3} - 11 \lambda^{2} - 32 \lambda - 256\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right) \left(\lambda + 5\right)^{2} \left(\lambda - 4\right) \left(\lambda^{3} - 11 \lambda^{2} - 32 \lambda - 256\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 4 & 4 & 8 & 6 & 3 & 3 & 12 & 3 & 3 & 12 & 6 & 12 & 8 & 8 & 16 & 8 & 8 & 20 \\ 1 & -1 & 1 & 0 & 0 & 1 & 2 & 4 & 4 & 6 & 0 & 0 & 2 & 0 & 2 & 3 & 3 & 6 & 6 & 0 & 8 \\ 1 & 1 & -1 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 4 & 4 & 6 & 0 & 2 & 3 & 3 & 6 & 0 & 6 & 8 \\ 1 & 0 & 0 & -1 & 1 & 1 & 0 & 4 & 0 & 2 & 4 & 0 & 2 & 2 & 6 & 6 & 0 & 6 & 3 & 3 & 8 \\ 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & 4 & 2 & 0 & 4 & 2 & 2 & 2 & 0 & 6 & 6 & 3 & 3 & 8 \\ 1 & 1 & 1 & 1 & 1 & -1 & 2 & 0 & 0 & 2 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & -2 & -4 & -2 & 0 & -2 & -1 & 0 & -2 & -1 & -2 & -4 & 4 & 0 & 0 & 0 & 0 & 6 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 2 & 0 & 2 & 2 & 0 & 3 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & -2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 3 \\ 0 & 1 & 0 & 0 & 1 & 3 & 1 & 1 & 2 & -1 & 0 & 1 & 2 & 1 & 2 & 0 & 2 & 2 & 2 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & -2 & 1 & 1 & 0 & 1 & 2 & 0 & 2 & 0 & 2 & 3 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 2 & 2 & 0 & 2 & 3 \\ 0 & 0 & 1 & 0 & 1 & 3 & 1 & 0 & 1 & 2 & 1 & 2 & -1 & 1 & 2 & 0 & 2 & 2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & 3 & 2 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 3 & 0 & 4 & 0 & 4 & 4 & 6 \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 0 & -3 & 2 & -2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & -\frac{1}{2} & 0 & \frac{3}{2} & -\frac{1}{2} & 0 & -1 & -1 & -2 & 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -\frac{1}{2} & \frac{3}{2} & 0 & -\frac{1}{2} & \frac{3}{2} & 0 & -1 & -3 & 0 & -2 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{1}{2} & \frac{1}{2} & 3 & \frac{1}{2} & \frac{1}{2} & 3 & 1 & 3 & 2 & 2 & -1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 2 & 3 & 1 & 1 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 3 & 1 & 1 & 0 & 2 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$