Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-262-58-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ3P3 × ℙ3P3
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_8 (marked)node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(w3 + w8) + E^(w4 + w5) + E^(w1 + w5) + E^(w2 + w8)
Bundle
O_2(1) box O_4(1) + Q_2 box O_3(1) + O_1(1) box O_3(1) + Q_1 box O_4(1)
ZL62 label
20203030.wRwIwBw6
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 58
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 262
$\chi(X, \mathrm{T}_X)$ = 2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 262
$\mathrm{c}_1^2\mathrm{c}_2$ = 160
$\mathrm{c}_2^2$ = 101
$\mathrm{c}_1\mathrm{c}_3$ = 58
$\mathrm{c}_4$ = 19
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 9 0 0
0 0 0 0
0 4 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 + 38 0 0 x1 - 20 58
$\mathrm{HH}^{ 4 }$ = 58 0 0 0 0 58
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 53t + 154t^{2} + 53t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(189)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(266)

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) \left( 344t^{3} + 72t^{2} - 2t - 1 \right) \left( 1991t^{7} + 2725t^{6} + 1457t^{5} + 529t^{4} + 186t^{3} + 37t^{2} + 7t + 1 \right) \left( 4147164042654265344t^{21} + 19001360289950954496t^{20} + 36791385038779436288t^{19} + 34473579776581573120t^{18} + 12500071823743902720t^{17} - 7456014197551222464t^{16} - 13473707230059480848t^{15} - 10066624109884305024t^{14} - 5049130136235515830t^{13} - 1857553007238117468t^{12} - 504018081378453761t^{11} - 97912732355040627t^{10} - 12989030883065937t^{9} - 1145156015914019t^{8} - 83504548542295t^{7} - 9336315389625t^{6} - 1177240823685t^{5} - 100915815729t^{4} - 5055988442t^{3} - 148258939t^{2} - 2450408t - 28820 \right)D^{6} + \left( 119297188141743231949012992t^{32} + 773297592834755531535482880t^{31} + 2265711564283213549314043904t^{30} + 3826246258472558347944318976t^{29} + 4000131884752255641742289920t^{28} + 2403971002616243072781647872t^{27} + 203898858492166207853752320t^{26} - 1251120676356455313210679424t^{25} - 1579563089632374239624000288t^{24} - 1212879241341211076642082480t^{23} - 699194988903113642612634296t^{22} - 322810758257869576507699048t^{21} - 122698361353095557513285034t^{20} - 38906484454648945249372978t^{19} - 10355519386696097450800565t^{18} - 2317426180779761660504243t^{17} - 435942717284402692461770t^{16} - 69377200506083702591216t^{15} - 9676995976430848523302t^{14} - 1298565687065454176635t^{13} - 185817275543778323955t^{12} - 27305498485161375572t^{11} - 3535601988986102524t^{10} - 359095265066196855t^{9} - 28027385948247345t^{8} - 1928238815997496t^{7} - 147376650326307t^{6} - 11440465796579t^{5} - 678941095424t^{4} - 25498264330t^{3} - 566600028t^{2} - 6717184t - 57640 \right)D^{5} + \left( 994143234514526932908441600t^{32} + 6302880818308051887056486400t^{31} + 17927476311593269381117243392t^{30} + 28886242681683865634786318336t^{29} + 27888839684503242676729185280t^{28} + 13662511751069520144239906816t^{27} - 3314087468144843506378131456t^{26} - 12943511014588799923943018624t^{25} - 13617171894844219044935197792t^{24} - 9581598545658990947346340688t^{23} - 5158882301631299649276688936t^{22} - 2226364242245670005339520952t^{21} - 784899955781827779853341390t^{20} - 227787127803122685157367606t^{19} - 54521522884376531538477919t^{18} - 10733216841809167895342270t^{17} - 1722387172028193169509143t^{16} - 221467905563993330786614t^{15} - 22246645482061788407190t^{14} - 1691728736208627493075t^{13} - 91587549028327031322t^{12} - 2354458406661878735t^{11} + 93986637135180436t^{10} - 33212146124082890t^{9} - 14701276467866919t^{8} - 2072303393958774t^{7} - 127212912558824t^{6} - 949492533876t^{5} + 355620595852t^{4} + 21035692825t^{3} + 531135085t^{2} + 6037556t + 28820 \right)D^{4} + t \left( 4175401584961013118215454720t^{31} + 26005922334354612035697868800t^{30} + 72154222064721940120475176960t^{29} + 111579468735233637538371430400t^{28} + 100032410000779231945535005696t^{27} + 38391921897226432618429028352t^{26} - 27699955492683803112472137728t^{25} - 60229094451212411360259115392t^{24} - 57293426479239355740012033632t^{23} - 38106272556224056799444368272t^{22} - 19586563054994982030814434216t^{21} - 8058031974277876164951591480t^{20} - 2686124509257220941513295134t^{19} - 727428932578640533039989214t^{18} - 159831828493984017357079603t^{17} - 28425500108816066886919633t^{16} - 4090268367606112704422298t^{15} - 482324648809404941730698t^{14} - 49703492540984923991616t^{13} - 5291575329697222179775t^{12} - 660806287618757378195t^{11} - 83318429469998307298t^{10} - 8639024107475660060t^{9} - 680323642680680349t^{8} - 42036822954328167t^{7} - 2281307086080636t^{6} - 117438219283663t^{5} - 5230245623439t^{4} - 167105894730t^{3} - 3422206110t^{2} - 34893128t - 12760 \right)D^{3} + 2 t ^{2} \left( 4612824608147404968695169024t^{30} + 28336321646997909270560317440t^{29} + 77061427122299368200264257536t^{28} + 115135417778254890128540715008t^{27} + 96606480890411261051271099392t^{26} + 27571837549894978656317975552t^{25} - 40711819136312473609199375616t^{24} - 70248298375902926193065132928t^{23} - 62756575695922942853310637376t^{22} - 40181697417580348799865855872t^{21} - 19995520010965284661979805584t^{20} - 7947762950414664529108494072t^{19} - 2540800421420842673536264448t^{18} - 652137623197630543913021104t^{17} - 133763229554186909308739111t^{16} - 21866630991729703392387363t^{15} - 2873668350260566562163246t^{14} - 318108572474055517127078t^{13} - 33849663203877270175562t^{12} - 4074210194995893195537t^{11} - 528577434482963863905t^{10} - 60319624537830249376t^{9} - 5298615313296212918t^{8} - 344606278303498223t^{7} - 18031173176218527t^{6} - 949260992340790t^{5} - 54155249158989t^{4} - 2575582753359t^{3} - 82021902734t^{2} - 1590316150t - 13535968 \right)D^{2} + 4 t ^{2} \left( 2505240950976607870929272832t^{30} + 15232730705411126172603310080t^{29} + 40797805593689427977232850944t^{28} + 59322480362235701799461646336t^{27} + 47085812749265960049815283712t^{26} + 9391857213395969930988876800t^{25} - 25738501559532527235267326720t^{24} - 39323385125104339024326747904t^{23} - 33728648444666989224115201760t^{22} - 21043486013091786078610307376t^{21} - 10237298266541672429794738728t^{20} - 3969288425691619149495387888t^{19} - 1230135109995670125701066010t^{18} - 303110874922855903646755818t^{17} - 58934009927786490774172842t^{16} - 9014438845367836451022662t^{15} - 1105848043022276307073894t^{14} - 118727216055720404448820t^{13} - 13428717921413541683769t^{12} - 1748055141622424170615t^{11} - 221209616123184997361t^{10} - 22725472917909072651t^{9} - 1741457925939801594t^{8} - 97837067127806771t^{7} - 4644857995824295t^{6} - 258581819989973t^{5} - 16168420030002t^{4} - 762428000182t^{3} - 22759496180t^{2} - 400059988t - 2766720 \right)D + 16 t ^{2} \left( 255636831732306925605027840t^{30} + 1543549118767851328142745600t^{29} + 4090381820718321928921694208t^{28} + 5833408211475443700054441984t^{27} + 4440137629432698697623384064t^{26} + 586941038466648641064963072t^{25} - 2860433104540528468881355008t^{24} - 4081314182939140579603800576t^{23} - 3411961862040021757522507744t^{22} - 2093082751215297338229266288t^{21} - 1002970459761049372551491880t^{20} - 382322452455936784790361760t^{19} - 115934881782129871104831282t^{18} - 27744895709609320226926978t^{17} - 5188041852158492921093976t^{16} - 755532721560944916315988t^{15} - 88276854470135053904900t^{14} - 9391919966396271533346t^{13} - 1126804232282635576385t^{12} - 152088755673543379569t^{11} - 18393702024873271965t^{10} - 1719938189933402851t^{9} - 117994682553809384t^{8} - 5986513296585681t^{7} - 280241805223847t^{6} - 17391266252883t^{5} - 1134963213080t^{4} - 50984301208t^{3} - 1394535088t^{2} - 21669340t - 115280 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ3P3 × ℙ3P3

|λ| = 9.458063 λ ≈ 9.458063; multiplicity 1; Jordan blocks 1 λ ≈ -5.729032 - 1.883954i; multiplicity 1; Jordan blocks 1 λ ≈ -5.729032 + 1.883954i; multiplicity 1; Jordan blocks 1 λ ≈ -4.478697; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.6287158 - 3.649223i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.6287158 + 3.649223i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.3051486 - 3.47142i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.3051486 + 3.47142i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.584218 - 0.3999044i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.584218 + 0.3999044i; multiplicity 2; Jordan blocks 1, 1 2 2 2 2 2 2 2
Sector dimension
18
Spectral radius
$9.45806308$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.458063$11
$-5.729032 - 1.883954\mathrm{i}$11
$-5.729032 + 1.883954\mathrm{i}$11
$-4.478697$21, 1
$0.6287158 - 3.649223\mathrm{i}$21, 1
$0.6287158 + 3.649223\mathrm{i}$21, 1
$-0.3051486 - 3.47142\mathrm{i}$21, 1
$-0.3051486 + 3.47142\mathrm{i}$21, 1
$-2$11
$-1.584218 - 0.3999044\mathrm{i}$21, 1
$-1.584218 + 0.3999044\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{3} + 2 \lambda^{2} - 72 \lambda - 344\right) \left(\lambda^{7} + 7 \lambda^{6} + 37 \lambda^{5} + 186 \lambda^{4} + 529 \lambda^{3} + 1457 \lambda^{2} + 2725 \lambda + 1991\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{3} + 2 \lambda^{2} - 72 \lambda - 344\right) \left(\lambda^{7} + 7 \lambda^{6} + 37 \lambda^{5} + 186 \lambda^{4} + 529 \lambda^{3} + 1457 \lambda^{2} + 2725 \lambda + 1991\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 2 & 2 & 0 & 3 & 3 & 3 & 3 & 3 & 3 & 6 & 12 & 12 & 8 & 12 & 20 \\ 1 & -1 & 0 & 1 & 1 & 2 & 0 & 0 & -2 & 2 & 0 & 0 & 0 & 0 & 3 & 3 & 3 & 0 \\ 1 & 0 & -1 & 1 & 1 & 2 & 0 & 2 & 0 & 0 & 0 & -2 & 0 & 3 & 0 & -3 & 6 & 0 \\ 1 & 0 & 1 & -1 & 0 & 0 & 2 & 2 & 2 & 0 & 2 & 2 & 2 & 0 & 3 & 3 & 0 & 8 \\ 1 & 1 & 0 & 0 & -1 & 0 & 2 & 0 & 2 & 2 & 2 & 2 & 2 & 3 & 0 & 0 & 0 & 8 \\ 0 & 1 & 1 & 0 & 0 & -2 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & -2 & -2 & -2 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & -2 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 2 & 2 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 6 \\ 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 2 & 2 & 0 & 4 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 2 & 2 & 0 & 6 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0 & 2 & 1 & 0 & 0 & 0 & -1 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 & 2 & 3 & 5 & -7 & -1 & -2 & -6 & -1 & 0 & -1 & 4 & -4 \\ 0 & 0 & 0 & 0 & 0 & -1 & -2 & -2 & -5 & 7 & 4 & 4 & 6 & 2 & -1 & 0 & -3 & 4 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & -2 & 4 & 2 & 1 & 6 & 1 & 0 & 1 & -3 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{5}{2} & \frac{3}{2} & 3 & 0\end{pmatrix}\notag\end{equation}$$