Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-60-19-b

·

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) + 2 copies of E^(w1 + w3)
Bundle
2 copies of O_2(2) + 2 copies of O_1(1) box O_2(1)
ZL62 label
20600.AAw1w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 19
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 60
$\chi(X, \mathrm{T}_X)$ = -34

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 60
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 148
$\mathrm{c}_1\mathrm{c}_3$ = 0
$\mathrm{c}_4$ = 48
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 38 2 0
0 0 0 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 + 34 0 x0 + 34 x2
$\mathrm{HH}^{ 3 }$ = x1 + 2 0 0 x1 2
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 15 0 0 x1 - x2 - 4 0 19
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 14t + 30t^{2} + 14t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(678)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(678)

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( 5t + 1 \right) ^{4} \left( 3888t^{3} + 36t - 1 \right) \left( 23003989104204t^{9} + 26233375996800t^{8} + 12138958256760t^{7} + 2866308532205t^{6} + 365377020055t^{5} + 26354393219t^{4} + 1349324915t^{3} + 54047785t^{2} + 152150t + 150 \right)D^{6} + 3 \left( 4t + 1 \right) \left( 5t + 1 \right) ^{3} \left( 12521531349200321280t^{14} + 17970450486199118784t^{13} + 11078177052338014464t^{12} + 3817958112699604992t^{11} + 811481886823926060t^{10} + 112633979693169948t^{9} + 10800817976328344t^{8} + 762114258658364t^{7} + 41896794416387t^{6} + 1949240442685t^{5} + 71858015763t^{4} + 1409881005t^{3} + 70772930t^{2} + 152050t + 100 \right)D^{5} + \left( 5t + 1 \right) ^{2} \left( 6260765674600160640000t^{16} + 11025133859261950963200t^{15} + 8575831328364753754896t^{14} + 3863709296943400141776t^{13} + 1115826484545003700944t^{12} + 217213813248165166428t^{11} + 29463790082113749400t^{10} + 2882657800778883224t^{9} + 212218358571516061t^{8} + 11992209461032919t^{7} + 479147939284888t^{6} + 9834364811450t^{5} - 42886121552t^{4} - 5777257195t^{3} - 168233885t^{2} - 311550t - 150 \right)D^{4} + t \left( 5t + 1 \right) \left( 131476079166603373440000t^{16} + 244995333910594913088000t^{15} + 203420927415524907535200t^{14} + 98885097341004262931136t^{13} + 31209733618900002167472t^{12} + 6734456894024714810988t^{11} + 1026154257474475601988t^{10} + 113731954956412305084t^{9} + 9506669006217434224t^{8} + 618342844210274747t^{7} + 30652245027390293t^{6} + 1026176413629158t^{5} + 17810958769174t^{4} + 48050277493t^{3} - 1151818205t^{2} - 828310t + 950 \right)D^{3} + 2 t ^{2} \left( 726248818253618634240000t^{16} + 1444345048545373130112000t^{15} + 1290596061103219504380600t^{14} + 682143885601783754643720t^{13} + 236974679598211723692936t^{12} + 57058207860155788000800t^{11} + 9832621635821583550688t^{10} + 1244403442782917570664t^{9} + 118991962193239460604t^{8} + 8845914901084307523t^{7} + 511825326050925300t^{6} + 21472092202556598t^{5} + 547182705406284t^{4} + 5562766667334t^{3} - 18668531148t^{2} + 133208195t + 419220 \right)D^{2} + 16 t ^{2} \left( 98607059374952530080000t^{16} + 190713103959841929072000t^{15} + 165202690210748382080250t^{14} + 84305784566071885411350t^{13} + 28150233757015233340242t^{12} + 6486777114566397405489t^{11} + 1066693278814602170988t^{10} + 128864307942190458240t^{9} + 11817943205935120420t^{8} + 845974965555312742t^{7} + 46541624551580604t^{6} + 1769695183965968t^{5} + 36259671066140t^{4} + 145696436970t^{3} - 3619588910t^{2} + 17544100t + 42300 \right)D + 32 t ^{2} \left( 20123889668357659200000t^{16} + 38177321021070086880000t^{15} + 32356541199299047602300t^{14} + 16102382199164269141860t^{13} + 5223921734461281099948t^{12} + 1165427495035234831242t^{11} + 185104036784103556824t^{10} + 21609317999589119160t^{9} + 1922652142037789100t^{8} + 133779383507338143t^{7} + 7055468598344902t^{6} + 246106011021884t^{5} + 4085606970910t^{4} - 6806116550t^{3} - 621669670t^{2} + 895900t + 3300 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ6P6

|λ| = 38.60834 λ ≈ 38.60834; multiplicity 1; Jordan blocks 1 λ ≈ -1.304169 - 9.950014i; multiplicity 1; Jordan blocks 1 λ ≈ -1.304169 + 9.950014i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 2, 2 λ ≈ -4; multiplicity 2; Jordan blocks 2 4 2
Sector dimension
9
Spectral radius
$38.6083389$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$38.60834$11
$-1.304169 - 9.950014\mathrm{i}$11
$-1.304169 + 9.950014\mathrm{i}$11
$-5$42, 2
$-4$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{4} \left(\lambda^{3} - 36 \lambda^{2} - 3888\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{2} \left(\lambda^{3} - 36 \lambda^{2} - 3888\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 64 & 600 & 180 & 924 & 9568 & 3424 & 52960 \\ 1 & -1 & 8 & 56 & 8 & 120 & 864 & 288 & 5440 \\ 1 & 2 & 4 & 64 & 24 & 96 & 1200 & 456 & 6848 \\ 0 & 1 & 1 & 3 & 0 & 16 & 72 & 16 & 480 \\ 0 & 1 & 0 & 8 & -1 & 4 & 112 & 40 & 528 \\ 0 & 0 & 1 & 2 & 1 & 0 & 56 & 24 & 360 \\ 0 & 0 & 0 & 1 & 0 & 3 & 3 & 2 & 48 \\ 0 & 0 & 0 & 1 & 1 & -3 & 10 & 0 & 32 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$