Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-163-40-a

·

This is a Fano fourfold of K3 type, appearing as case C-4 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ3P3 × ℙ5P5
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_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w1 + 2w4) + E^(w3 + w4)
Bundle
O_1(1) box O_2(2) + Q_1 box O_2(1)
ZL62 label
3050.wDx321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 40
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 163
$\chi(X, \mathrm{T}_X)$ = -24

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 163
$\mathrm{c}_1^2\mathrm{c}_2$ = 142
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 22
$\mathrm{c}_4$ = 31
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 23 1 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 + 24 0 x0 + 24 x2
$\mathrm{HH}^{ 3 }$ = x1 + 32 0 0 x1 32
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 38 0 0 x1 - x2 - 2 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 35t + 91t^{2} + 35t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(131)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(174)

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( 2t + 1 \right) ^{2} \left( 88191t^{7} + 109740t^{6} + 55760t^{5} + 11776t^{4} + 829t^{3} + 4t^{2} + 2t - 1 \right) \left( 41056016251223826t^{13} + 129902102094576312t^{12} + 174897624092328735t^{11} + 133372210701833275t^{10} + 66125526686220176t^{9} + 23416239686167416t^{8} + 6199513186549212t^{7} + 1194358124363730t^{6} + 153144204063708t^{5} + 11884882271325t^{4} + 573331514895t^{3} + 20493808899t^{2} + 52780400t + 945400 \right)D^{6} + \left( 2t + 1 \right) \left( 152072387426890578428172t^{21} + 704329226851870435712166t^{20} + 1483176176602515772196598t^{19} + 1890814099887447281315703t^{18} + 1643803214352709913022576t^{17} + 1042742583232096756293730t^{16} + 504644559853486507087442t^{15} + 191899262503269797034177t^{14} + 58180054000226741738838t^{13} + 14038998594681010601376t^{12} + 2652341093574476660883t^{11} + 382756967523762295151t^{10} + 41310157452421579684t^{9} + 3361303268340450360t^{8} + 226722574720865180t^{7} + 15608364461821981t^{6} + 1148752423782828t^{5} + 68289705005142t^{4} + 2713606056483t^{3} + 81613497996t^{2} + 150778000t + 1890800 \right)D^{5} + \left( 2534539790448176307136200t^{22} + 12444788019430964671852680t^{21} + 27850711617747935651357550t^{20} + 37844937830782824475255992t^{19} + 35191154424347932368960249t^{18} + 23976488656171066272598271t^{17} + 12524116102592697007580166t^{16} + 5171318502235064840078396t^{15} + 1716499702892609763284091t^{14} + 459017126120437009375141t^{13} + 97789140904160690151816t^{12} + 16260243219047402968882t^{11} + 2059512998434598182580t^{10} + 194083329288052272109t^{9} + 13254178717170870790t^{8} + 600068152979968509t^{7} + 8138400149019481t^{6} - 1192560698133771t^{5} - 86557244368927t^{4} - 2629486305429t^{3} - 71122738201t^{2} - 102706040t - 945400 \right)D^{4} + t \left( 10645067119882340489972040t^{21} + 50415812014639870167951240t^{20} + 108283231404824782914894870t^{19} + 140674834863302990296739580t^{18} + 124868717548406497742162361t^{17} + 81307308987841445446959776t^{16} + 40724302565266398096144418t^{15} + 16179354162892263260879680t^{14} + 5171184829360557050909691t^{13} + 1326805550301097448575790t^{12} + 269312867176067936702042t^{11} + 42364064477934107867173t^{10} + 5066113310778167786157t^{9} + 455870984691799967760t^{8} + 30941297829616876420t^{7} + 1569033706073089244t^{6} + 53501656995383699t^{5} + 879549166627356t^{4} + 2554080234418t^{3} - 33440632775t^{2} + 343258760t + 1305000 \right)D^{3} + 2 t ^{2} \left( 11760264627679538065111968t^{20} + 54131835959567822242427904t^{19} + 112497190384836798510944328t^{18} + 140944574012359385896925112t^{17} + 120511475891571322887782889t^{16} + 75684311488205748930900034t^{15} + 36667438524136319133568492t^{14} + 14122444419521704155350868t^{13} + 4371819892858543403452443t^{12} + 1081088124518996115021722t^{11} + 209927015280008649216933t^{10} + 31371340299524646100361t^{9} + 3552473753993104429731t^{8} + 303677369577528673254t^{7} + 19709202831258466523t^{6} + 947141906504351576t^{5} + 29252128757408211t^{4} + 433924188082431t^{3} + 3666634841398t^{2} + 47476033059t + 168799540 \right)D^{2} + 8 t ^{2} \left( 3193520135964702146991612t^{20} + 14388034305028820699430252t^{19} + 29159592873411289245996141t^{18} + 35527310325040158972266922t^{17} + 29511959700631359347113071t^{16} + 18026303169312571953350788t^{15} + 8512869599154970726425084t^{14} + 3199831800213749014433430t^{13} + 964941855061844230947159t^{12} + 231252542881908393408789t^{11} + 43229072173518381233055t^{10} + 6184272175881744475679t^{9} + 669147097664061041130t^{8} + 54836324918799759087t^{7} + 3419480525040088376t^{6} + 154567391966060652t^{5} + 4179820193721648t^{4} + 49793512114644t^{3} + 568928134770t^{2} + 8001636360t + 20798800 \right)D + 16 t ^{2} \left( 651738803258102478977880t^{20} + 2893376721180042687874680t^{19} + 5761888720844931984159810t^{18} + 6882841781992322971362108t^{17} + 5600929694227807893770520t^{16} + 3353774299816647087629556t^{15} + 1554921290337118449076125t^{14} + 574082942826067896613674t^{13} + 169693555143006748421458t^{12} + 39679555678266506696660t^{11} + 7196985836231479404479t^{10} + 994598664002241638362t^{9} + 103844526276619659594t^{8} + 8236156384337784447t^{7} + 497182957915039698t^{6} + 21263343932999040t^{5} + 505950279929742t^{4} + 4621739059022t^{3} + 62198415250t^{2} + 712434880t + 1890800 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ5P5

|λ| = 13.52071 λ ≈ 13.52071; multiplicity 1; Jordan blocks 1 λ ≈ -0.2139247 - 10.03787i; multiplicity 1; Jordan blocks 1 λ ≈ -0.2139247 + 10.03787i; multiplicity 1; Jordan blocks 1 λ ≈ -3.91263 - 1.21911i; multiplicity 1; Jordan blocks 1 λ ≈ -3.91263 + 1.21911i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ -1.633801 - 1.087845i; multiplicity 1; Jordan blocks 1 λ ≈ -1.633801 + 1.087845i; multiplicity 1; Jordan blocks 1 2
Sector dimension
9
Spectral radius
$13.5207111$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.52071$11
$-0.2139247 - 10.03787\mathrm{i}$11
$-0.2139247 + 10.03787\mathrm{i}$11
$-3.91263 - 1.21911\mathrm{i}$11
$-3.91263 + 1.21911\mathrm{i}$11
$-2$22
$-1.633801 - 1.087845\mathrm{i}$11
$-1.633801 + 1.087845\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{7} - 2 \lambda^{6} - 4 \lambda^{5} - 829 \lambda^{4} - 11776 \lambda^{3} - 55760 \lambda^{2} - 109740 \lambda - 88191\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{7} - 2 \lambda^{6} - 4 \lambda^{5} - 829 \lambda^{4} - 11776 \lambda^{3} - 55760 \lambda^{2} - 109740 \lambda - 88191\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 18 & 18 & 18 & 168 & 144 & 240 \\ 2 & 0 & 5 & 4 & -2 & 6 & 0 & 3 & 72 \\ 1 & 0 & 0 & 0 & 2 & 0 & 45 & 42 & 96 \\ 0 & 1 & 2 & 0 & 0 & 3 & 0 & -2 & 0 \\ 0 & 2 & 0 & 5 & 0 & 6 & 6 & 4 & 0 \\ 0 & 0 & 1 & 0 & 0 & -2 & 0 & 2 & 18 \\ 0 & 0 & 0 & 1 & -\frac{8}{3} & 3 & -10 & -\frac{20}{3} & -8 \\ 0 & 0 & 0 & 2 & 5 & 0 & 15 & 10 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{7}{3} & 0\end{pmatrix}\notag\end{equation}$$