Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-401-84-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ4P4
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_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w4 + w5) + E^(w2 + w5)
Bundle
Q_2 box O_3(1) + Q_1 box O_3(1)
ZL62 label
202040.wIw9
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 84
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 401
$\chi(X, \mathrm{T}_X)$ = 12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 401
$\mathrm{c}_1^2\mathrm{c}_2$ = 194
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 58
$\mathrm{c}_4$ = 13
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 5 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 - 12 0 x0 - 12 x1
$\mathrm{HH}^{ 3 }$ = x1 + 60 0 0 x1 - 60 120
$\mathrm{HH}^{ 4 }$ = 84 0 0 0 0 84
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 79t + 241t^{2} + 79t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(16)
Coates et al. (2020) Obro(4,114)
Coates–Kasprzyk–Prince (2015) CKP(21)

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( 43t^{4} + 36t^{3} + 8t^{2} + t + 1 \right) \left( 3341t^{5} + 450t^{4} + 250t^{3} - t^{2} - 2t - 1 \right) \left( 15067926119125554t^{13} + 15194330816420088t^{12} + 8188278143345637t^{11} + 7914513545454165t^{10} + 7075620636459012t^{9} + 1813198970425094t^{8} - 11907596529706t^{7} - 12239183675778t^{6} - 1051958607061t^{5} + 411151483477t^{4} - 28072769775t^{3} + 2462836281t^{2} + 31481972t + 837288 \right)D^{6} + \left( 45458772871090623750342t^{22} + 83788851546999318499236t^{21} + 75758145086085638778153t^{20} + 62585547243216628565682t^{19} + 56949710545060626831360t^{18} + 36274310439002098789220t^{17} + 13823697027735049372023t^{16} + 3495080577865332067197t^{15} + 725185395649243424230t^{14} + 106456515067724312845t^{13} + 4779194431533024284t^{12} + 470307166627577707t^{11} + 235168978009453923t^{10} + 122687226920402969t^{9} + 17731210481656230t^{8} + 300404308325534t^{7} - 134759242027064t^{6} + 232614846934t^{5} + 2336906943993t^{4} - 140445899975t^{3} + 9701638200t^{2} + 89422188t + 1674576 \right)D^{5} + \left( 378823107259088531252850t^{22} + 656689537150616058767820t^{21} + 563766276906046533474879t^{20} + 479950854890510060589123t^{19} + 461758132341188593913286t^{18} + 283578749348615923992396t^{17} + 95921995410102779596279t^{16} + 19120717937356995011368t^{15} + 2651457893032502773601t^{14} + 219641985886414853669t^{13} - 22606985076318272421t^{12} - 5604803238516907149t^{11} - 374042145856355594t^{10} - 154195272259281176t^{9} - 51087556934731089t^{8} + 1787714305229298t^{7} + 290314994680192t^{6} + 51167417570156t^{5} - 2460112156681t^{4} + 180808103270t^{3} - 9445046967t^{2} - 106859272t - 837288 \right)D^{4} + t \left( 1591057050488171831261970t^{21} + 2620978111087139182029660t^{20} + 2150725669961073744229695t^{19} + 1898384838669614739699594t^{18} + 1921117710220082235347328t^{17} + 1134137951393759970348112t^{16} + 341602533431016652994529t^{15} + 53396644152319476035499t^{14} + 4029505897619374537178t^{13} - 75625004910461464051t^{12} - 67176195671334959892t^{11} - 11097001051518547651t^{10} + 405588770900827749t^{9} + 261018397121415881t^{8} - 23176527053086852t^{7} + 2490075868519216t^{6} + 99428604790426t^{5} + 28158059989536t^{4} + 593039210865t^{3} + 16346190623t^{2} + 59338280t + 77256 \right)D^{3} + 2 t ^{2} \left( 1757739217682170785013224t^{20} + 2779659459735234396332076t^{19} + 2196157874888138089343817t^{18} + 2010130151211599965713828t^{17} + 2120067762072697342086234t^{16} + 1203455345047796556096430t^{15} + 325095260406870546811295t^{14} + 39807811875543381317795t^{13} + 600467289324346268265t^{12} - 437394253207158460709t^{11} - 21125891581583838974t^{10} - 2415374182913275789t^{9} + 185822894748772038t^{8} + 134355769613502734t^{7} - 24496129315403653t^{6} + 1590048719200489t^{5} + 149399895920301t^{4} + 15024737081085t^{3} + 354924129483t^{2} + 10048889661t + 18690610 \right)D^{2} + 8 t ^{2} \left( 477317115146451549378591t^{20} + 731757887041020285656268t^{19} + 561072604399083376733304t^{18} + 530166566817444999340404t^{17} + 576932877608445942099063t^{16} + 316329980767987320183393t^{15} + 77645991803826830468649t^{14} + 7488882880765982822169t^{13} - 336709293258915977172t^{12} - 136690650401322634869t^{11} + 4529429475294172784t^{10} + 544171117351838938t^{9} - 62619458792029819t^{8} + 11277163682552135t^{7} - 5488583561666271t^{6} + 359685040349793t^{5} + 38008957845239t^{4} + 2171161702793t^{3} + 60855146398t^{2} + 1577316032t + 1116384 \right)D + 144 t ^{3} \left( 10823517350259672321510t^{19} + 16239825450977969101680t^{18} + 12186638410984724498394t^{17} + 11796918699139946915364t^{16} + 13116310405914351737324t^{15} + 7004613853909669233340t^{14} + 1594912312090909888451t^{13} + 123615944855309283422t^{12} - 12991178387032087446t^{11} - 2958401854679353302t^{10} + 242146045188362291t^{9} + 22072157054396477t^{8} - 3491764084824269t^{7} - 62915129930599t^{6} - 100021934075883t^{5} + 7881739165659t^{4} + 757701558360t^{3} + 30023924838t^{2} + 975452490t + 22606776 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ4P4

|λ| = 6.670902 λ ≈ 6.670902; multiplicity 1; Jordan blocks 1 λ ≈ -3.794084 - 4.769462i; multiplicity 1; Jordan blocks 1 λ ≈ -3.794084 + 4.769462i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5413668 - 3.631922i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5413668 + 3.631922i; multiplicity 1; Jordan blocks 1 λ ≈ 1.206542 - 3.383479i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.206542 + 3.383479i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.706542 - 0.6481502i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.706542 + 0.6481502i; multiplicity 2; Jordan blocks 1, 1 2 2 2 2
Sector dimension
13
Spectral radius
$6.67090244$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.670902$11
$-3.794084 - 4.769462\mathrm{i}$11
$-3.794084 + 4.769462\mathrm{i}$11
$-0.5413668 - 3.631922\mathrm{i}$11
$-0.5413668 + 3.631922\mathrm{i}$11
$1.206542 - 3.383479\mathrm{i}$21, 1
$1.206542 + 3.383479\mathrm{i}$21, 1
$-1.706542 - 0.6481502\mathrm{i}$21, 1
$-1.706542 + 0.6481502\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{4} + \lambda^{3} + 8 \lambda^{2} + 36 \lambda + 43\right)^{2} \left(\lambda^{5} + 2 \lambda^{4} + \lambda^{3} - 250 \lambda^{2} - 450 \lambda - 3341\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{4} + \lambda^{3} + 8 \lambda^{2} + 36 \lambda + 43\right) \left(\lambda^{5} + 2 \lambda^{4} + \lambda^{3} - 250 \lambda^{2} - 450 \lambda - 3341\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 3 & 3 & 3 & 3 & 0 & 0 & 0 & 5 \\ 2 & 0 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 3 & 3 & 0 & 0 \\ 2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -2 & 3 & 0 & 3 & 0 \\ 1 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 2 & -1 & 0 & 1 & 0 & 1 & 0 & 0 & -2 & -2 & 0 \\ 0 & 1 & 0 & 3 & 0 & -1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 1 & 3 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 3 & 0 & 3 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 3 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 3 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$