Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-352-75-b

·

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

Varieties in this family can be blown up to

Defining data
Ambient
ℙ2P2 × Gr(2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
2 copies of E^(w4) + E^(w2 + w3)
Bundle
2 copies of O_2(1) + Q_1 box U_2^v
ZL62 label
2051.33w6
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 75
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 352
$\chi(X, \mathrm{T}_X)$ = 6

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 352
$\mathrm{c}_1^2\mathrm{c}_2$ = 184
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 10
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 4 0 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 + 36 0 x0 - 6 42
$\mathrm{HH}^{ 3 }$ = 102 0 0 0 102
$\mathrm{HH}^{ 4 }$ = 75 0 0 0 0 75
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 70t + 210t^{2} + 70t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(142)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( 3299t^{10} - 23328t^{9} + 3456t^{8} - 139t^{7} + 3464t^{6} - 6272t^{5} + 849t^{4} - 32t^{3} - 48t^{2} - t + 1 \right) \left( 293519627851776t^{16} - 190080841712256t^{15} + 283963672680768t^{14} - 1289852940216320t^{13} + 3043283973025876t^{12} - 3172483541717360t^{11} + 1003695855457817t^{10} - 560727319667384t^{9} + 340209838324368t^{8} - 51432392830091t^{7} - 56755722938460t^{6} + 15057616944048t^{5} - 2091824695680t^{4} - 334356926416t^{3} + 15098768880t^{2} - 860829184t - 1622016 \right)D^{6} + \left( 20334746297943189504t^{26} - 130198527264738034944t^{25} + 114549267190466369088t^{24} - 237551247464057880576t^{23} + 872857282578779191580t^{22} - 1840534386242729571296t^{21} + 1907044339688171180169t^{20} - 833874522824055630124t^{19} + 545678728375105062704t^{18} - 449724436924360659515t^{17} + 211493770851334228992t^{16} - 40334376807468844624t^{15} + 32365155422256209469t^{14} - 21653631602302576568t^{13} + 1897661890325679620t^{12} + 4388190139777344415t^{11} - 1555957323240645225t^{10} + 261473443173989184t^{9} + 5669477980444750t^{8} - 3753577825946750t^{7} + 962911430398028t^{6} - 106617450825216t^{5} + 12002306378720t^{4} + 2186986671312t^{3} - 75104560560t^{2} + 3444938752t + 4866048 \right)D^{5} + \left( 169456219149526579200t^{26} - 906575548424175694080t^{25} + 823722982117663925952t^{24} - 1931948777395243019776t^{23} + 7192153301586398174564t^{22} - 15242612618195300071728t^{21} + 15821212193031962848223t^{20} - 6738450362339144422908t^{19} + 4022789332401094533840t^{18} - 2682754507375637006868t^{17} + 694049315210985043892t^{16} + 219205959836629460512t^{15} - 25852568025132296946t^{14} - 37904224478522223524t^{13} - 4163954108114873268t^{12} + 20766815626466937140t^{11} - 6713178355740783343t^{10} + 962888342336347944t^{9} + 115555765028347875t^{8} - 17603000331285207t^{7} - 410069114679520t^{6} + 193439372225152t^{5} - 25374145588336t^{4} - 2622072261232t^{3} + 72375702928t^{2} - 3508473856t - 3244032 \right)D^{4} + t \left( 711716120428011632640t^{25} - 3218857113401021744640t^{24} + 3026419506718910078400t^{23} - 7926772244687492631552t^{22} + 29716218332323778198132t^{21} - 62779547592940432681216t^{20} + 64843901278639822497775t^{19} - 27573542364223482690196t^{18} + 16613592295073465460080t^{17} - 10489467558750928622595t^{16} + 1788131782950360081368t^{15} + 1933245182316597670048t^{14} - 677346548059760736059t^{13} + 91864207620283473432t^{12} - 34525537206765848884t^{11} + 49930577566693759943t^{10} - 15605344176131867279t^{9} + 2014530784599796352t^{8} + 408512096913861904t^{7} - 47464134401220664t^{6} + 2561059077598540t^{5} - 188971383627200t^{4} + 6623374192000t^{3} + 203829561152t^{2} + 807235632t - 4608 \right)D^{3} + 2 t ^{2} \left( 786276856853803327488t^{24} - 3058428174226168688064t^{23} + 2976434684500206301632t^{22} - 8578387324997962621184t^{21} + 32158629679150892527696t^{20} - 67235785277543265565360t^{19} + 68731052074573299154067t^{18} - 29450736836681769995400t^{17} + 18511260531420142187240t^{16} - 11847092560405498105851t^{15} + 1788037605471658441618t^{14} + 2793985543977416014192t^{13} - 1079871465387060766323t^{12} + 213667658938150387066t^{11} - 29823368062597020712t^{10} + 30899410878553482639t^{9} - 9794566051953717127t^{8} + 1080729195092080360t^{7} + 356498355806041640t^{6} - 36811253851137960t^{5} + 2062018982985402t^{4} - 88555853106616t^{3} + 4840696260800t^{2} + 40536184832t + 227803136 \right)D^{2} + 8 t ^{2} \left( 213514836128403489792t^{24} - 731491149368964864672t^{23} + 735096197294429246784t^{22} - 2290928481523256237952t^{21} + 8552209713875780428294t^{20} - 17636660331603635606760t^{19} + 17795516218938126963487t^{18} - 7704095079179650886414t^{17} + 5069501367189898305292t^{16} - 3331148941579127159312t^{15} + 496121109155202078098t^{14} + 888476918425309108744t^{13} - 348972735101445219905t^{12} + 71267217100611072614t^{11} - 4652098062447919130t^{10} + 4738035745372204960t^{9} - 1597258402419276800t^{8} + 139542945366515728t^{7} + 73189891860450920t^{6} - 7068662813899184t^{5} + 441775619572980t^{4} - 7525644171760t^{3} + 745457026464t^{2} + 2283586560t + 12976128 \right)D + 48 t ^{3} \left( 14524818784245135360t^{23} - 45209971586783972160t^{22} + 46640845735025849856t^{21} - 153988555602565262592t^{20} + 571983054815074533716t^{19} - 1164806773107293226576t^{18} + 1161277387365606642956t^{17} - 507441788874884142580t^{16} + 346523233671047542712t^{15} - 232712395879257830434t^{14} + 34800240198912350152t^{13} + 66731875460424574148t^{12} - 26163114353968812235t^{11} + 5272469883912392680t^{10} - 123388816147089940t^{9} + 186671968669762352t^{8} - 70652236757417620t^{7} + 4221822772304804t^{6} + 3859069308126364t^{5} - 355561638240688t^{4} + 24875546900928t^{3} - 16690652864t^{2} + 28584585504t + 106748928 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 6)

|λ| = 7.637783 λ ≈ 7.637783; multiplicity 1; Jordan blocks 1 λ ≈ -5.928852 - 3.121351i; multiplicity 1; Jordan blocks 1 λ ≈ -5.928852 + 3.121351i; multiplicity 1; Jordan blocks 1 λ ≈ 2.291565 - 3.398547i; multiplicity 1; Jordan blocks 1 λ ≈ 2.291565 + 3.398547i; multiplicity 1; Jordan blocks 1 λ ≈ 1.142688 - 0.9626318i; multiplicity 1; Jordan blocks 1 λ ≈ 1.142688 + 0.9626318i; multiplicity 1; Jordan blocks 1 λ ≈ -0.8965645 - 0.9852984i; multiplicity 1; Jordan blocks 1 λ ≈ -0.8965645 + 0.9852984i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1445435; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$7.63778299$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.637783$11
$-5.928852 - 3.121351\mathrm{i}$11
$-5.928852 + 3.121351\mathrm{i}$11
$2.291565 - 3.398547\mathrm{i}$11
$2.291565 + 3.398547\mathrm{i}$11
$1.142688 - 0.9626318\mathrm{i}$11
$1.142688 + 0.9626318\mathrm{i}$11
$-0.8965645 - 0.9852984\mathrm{i}$11
$-0.8965645 + 0.9852984\mathrm{i}$11
$0.1445435$11

Characteristic polynomial

$$\begin{equation}\lambda^{10} - \lambda^{9} - 48 \lambda^{8} - 32 \lambda^{7} + 849 \lambda^{6} - 6272 \lambda^{5} + 3464 \lambda^{4} - 139 \lambda^{3} + 3456 \lambda^{2} - 23328 \lambda + 3299\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{10} - \lambda^{9} - 48 \lambda^{8} - 32 \lambda^{7} + 849 \lambda^{6} - 6272 \lambda^{5} + 3464 \lambda^{4} - 139 \lambda^{3} + 3456 \lambda^{2} - 23328 \lambda + 3299\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 3 & 3 & 0 & 3 & 0 & 0 & 10 \\ 1 & 0 & 0 & 4 & 0 & 2 & 4 & -3 & -6 & 0 \\ 2 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 6 & 0 \\ 0 & 2 & 1 & 0 & -1 & 0 & 0 & 2 & 2 & -6 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 2 & 4 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & -4 & 7 & 6 & 0 & 0 & -8 \\ 0 & 0 & 0 & 4 & 4 & -2 & 1 & 0 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$