Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-74-22-a

·

This is a Fano fourfold of K3 type, appearing as case K3-40 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w2 + w4) + E^(w1 + 2w4)
Bundle
O_2(1) box O_3(1) + O_1(1) box O_3(2)
ZL62 label
1131.ex321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 22
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 74
$\chi(X, \mathrm{T}_X)$ = -29

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 74
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 42
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 32 1 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 + x2 + 29 0 x0 + 29 x2
$\mathrm{HH}^{ 3 }$ = x1 + 2 0 0 x1 2
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 14 0 0 x1 - x2 - 9 1 22
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 17t + 38t^{2} + 17t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(676)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(676)

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) \left( 6t + 1 \right) \left( 5t + 1 \right) ^{4} \left( 92t^{2} + 8t - 1 \right) \left( 1492t^{3} + 20t^{2} - 35t + 1 \right) \left( 5739605913230008262272t^{16} + 9636907397357147896960t^{15} + 7085035041414980294592t^{14} + 2979178597813353271168t^{13} + 788368027225781716888t^{12} + 136645658359081479320t^{11} + 15792399540107939572t^{10} + 1262399122747640796t^{9} + 83274220368366538t^{8} + 6417246668915884t^{7} + 528454213748919t^{6} + 31078456098637t^{5} + 1034408478880t^{4} + 15362902824t^{3} + 60623303t^{2} + 147871t - 1184 \right)D^{8} + \left( 5t + 1 \right) ^{3} \left( 3403474269437968649766016450560t^{24} + 7717807455683391446140299632640t^{23} + 7896627940325761296350212616192t^{22} + 4792675649771602124871328102400t^{21} + 1906593019639269013732937412608t^{20} + 517136852518065404180674861056t^{19} + 95716727152945110159134946816t^{18} + 11496401889502524943241742592t^{17} + 721286544162058627781329920t^{16} - 13947739118553229926012576t^{15} - 7569351908514841598211152t^{14} - 842068441225892546648904t^{13} - 76189903848624801120896t^{12} - 7682547286301793338876t^{11} - 678130877323827313576t^{10} - 40927203868630151516t^{9} - 1492323513860057592t^{8} - 16469578246631441t^{7} + 2328785407484878t^{6} + 213971816614095t^{5} + 8469185854412t^{4} + 132406205593t^{3} + 468920134t^{2} + 1094297t - 7104 \right)D^{7} + \left( 5t + 1 \right) ^{2} \left( 258096798765712622607256247500800t^{25} + 608840656422322970988974628003840t^{24} + 653577226290491625533208054031360t^{23} + 421198082196340420823826337399296t^{22} + 181102748372153985889405539768832t^{21} + 54604940261632649248054311104000t^{20} + 11803232772839969493147080737472t^{19} + 1835539784375314562320439119584t^{18} + 202999971094039763365727786592t^{17} + 15682369104031446305672643680t^{16} + 877353489549641170527699504t^{15} + 47972396700220963402761896t^{14} + 3166537177816213121199760t^{13} + 30346235613623626610292t^{12} - 35045153468755975212392t^{11} - 5065247885056063938836t^{10} - 373300536809234939530t^{9} - 17216669107852433213t^{8} - 608467043161406461t^{7} - 24706182325502567t^{6} - 1103123102992511t^{5} - 32734825105647t^{4} - 430381731435t^{3} - 1399343477t^{2} - 2329511t + 15392 \right)D^{6} + \left( 5t + 1 \right) \left( 10720943948729601246762951819264000t^{26} + 26457794180810845364372811056332800t^{25} + 29918321277762664322724913554191360t^{24} + 20492797952631191395152921713466880t^{23} + 9476849581638119592658034968775680t^{22} + 3123414792545651760325930906728960t^{21} + 755230101813393038226360111814080t^{20} + 136015371145901008591552196550432t^{19} + 18407784764144419074402469622336t^{18} + 1900638765067512576446104836800t^{17} + 157835394296041761701210094032t^{16} + 12054369980907491784898058408t^{15} + 962009692580912275309028560t^{14} + 71640411119321612824352332t^{13} + 3687631478994704413786208t^{12} + 56592406280373488426904t^{11} - 7533627615327034234916t^{10} - 648704775575102120951t^{9} - 23641489856201165726t^{8} - 234058194571693226t^{7} + 23482490716933766t^{6} + 1563642726088156t^{5} + 45267983615182t^{4} + 560575156378t^{3} + 1726768782t^{2} + 2105183t - 14208 \right)D^{5} + \left( 265293728731294993814573969786880000t^{27} + 687748640396209526072727287299584000t^{26} + 821653074960771868994637784537523200t^{25} + 598837180370988008879264092612533760t^{24} + 297266680452961931156404062167922176t^{23} + 106332148064199397364980995614041088t^{22} + 28295802650473968037420445866886976t^{21} + 5708975012476963333020838667261088t^{20} + 884790623474491149435525424493088t^{19} + 107017744505303381835640125311152t^{18} + 10467622187328842215578345043144t^{17} + 896437744423383378897833050560t^{16} + 74614198843689537334484277116t^{15} + 6085846587416716455162231280t^{14} + 424680553788486460570071618t^{13} + 20964354900154906940389708t^{12} + 501120691499703752754629t^{11} - 13024067454733261631781t^{10} - 1581245485140703265898t^{9} - 59652692848729431088t^{8} - 1182397312627237798t^{7} - 17977923084357306t^{6} - 558455346497430t^{5} - 17387971870292t^{4} - 235424453587t^{3} - 752770409t^{2} - 676146t + 4736 \right)D^{4} + 2 t \left( 397568338098722712900792796631040000t^{26} + 1006222581021019904188094834179584000t^{25} + 1171985970410432763123366820708236800t^{24} + 831465789276496504083040140888830720t^{23} + 401176472664538078125793183914935040t^{22} + 139302957533052261706151168812922112t^{21} + 35958464523683576919607943189487904t^{20} + 7039990050334467051411714100074256t^{19} + 1061666262306250952359825533178896t^{18} + 125961526828754691223022690867920t^{17} + 12304360678957531773965001640176t^{16} + 1080953610561209284172784252080t^{15} + 93510857980188211027211943576t^{14} + 7809035972547333891736571668t^{13} + 554650309609099038345287148t^{12} + 29307298821900980944981800t^{11} + 992482815717071484109110t^{10} + 13493138324268236255792t^{9} - 422586999109522456369t^{8} - 24500026192847065863t^{7} - 510637136355193303t^{6} - 6149572600708505t^{5} - 77534559158167t^{4} - 786570417621t^{3} - 3802816795t^{2} - 1949143t - 3440 \right)D^{3} + 4 t ^{2} \left( 348986106426293931236945249310720000t^{25} + 866213344099077013106037196486144000t^{24} + 988032586533524895078476127703347200t^{23} + 685355784227999815410991909303575040t^{22} + 322788343680795829118023861813585920t^{21} + 109240036094512172128057411894483200t^{20} + 27448456177451183389958513377411456t^{19} + 5228356751380677256755870467870304t^{18} + 768174497182233553962166071238656t^{17} + 89323071794669509416389930563392t^{16} + 8677582821907026174102120274472t^{15} + 774775369426626590867689983328t^{14} + 68702934432448334252061649012t^{13} + 5796112507802830370997754484t^{12} + 410803135726762524196420626t^{11} + 21884540234710660462382152t^{10} + 792678766586960185165345t^{9} + 16263800214772477848555t^{8} + 45433678021035280167t^{7} - 6373460111881356183t^{6} - 152736797160491187t^{5} - 1775246143409261t^{4} - 25679496008351t^{3} - 280580503707t^{2} - 1153122716t - 208384 \right)D^{2} + 16 t ^{3} \left( 80938872470071691952248078714880000t^{24} + 197873846122247739789899444381952000t^{23} + 222013518856804444875175556532230400t^{22} + 151256340497133427351137317705504640t^{21} + 69857268953963622392724654011169920t^{20} + 23146328755178403596914184178871296t^{19} + 5685954682192183194262476179670512t^{18} + 1057905742369199351007787228327464t^{17} + 151924683736547394761048863630424t^{16} + 17348659097851183090048851744880t^{15} + 1676035534206678892549506769920t^{14} + 151508034994328198452403908400t^{13} + 13671430036056975933512043540t^{12} + 1157743887726388436900920708t^{11} + 81355930405649142978021468t^{10} + 4300473775750774964417530t^{9} + 157890406010510001629934t^{8} + 3584790877159791930124t^{7} + 34605429283574516186t^{6} - 391767881600292978t^{5} - 14508995602489901t^{4} - 168585877587765t^{3} - 2206993177059t^{2} - 21900017871t - 80514516 \right)D + 192 t ^{3} \left( 2481699988131852140454386995200000t^{24} + 6001131218854668489961229057280000t^{23} + 6652792786073820302189360509536000t^{22} + 4472653409771016320432333514921600t^{21} + 2035589465818847451598691996191616t^{20} + 663693078411796923696512976579840t^{19} + 160214170117500945067046193855632t^{18} + 29263029942899359114715781544568t^{17} + 4126426260916228782637005690536t^{16} + 464368032446713787290119889872t^{15} + 44666292271132887504083948296t^{14} + 4077121855466807081869233520t^{13} + 372447358388960167772045072t^{12} + 31558237238336284165064624t^{11} + 2195921797351118090693382t^{10} + 114705687852021022470718t^{9} + 4197029204675170093163t^{8} + 98162540503889265543t^{7} + 1166170727594365557t^{6} - 2045782683985965t^{5} - 265744805282246t^{4} - 4036537320560t^{3} - 47428898004t^{2} - 343431940t - 943056 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × Gr(2, 4)

|λ| = 33.02657 λ ≈ 33.02657; multiplicity 1; Jordan blocks 1 λ ≈ 14.3923; multiplicity 1; Jordan blocks 1 λ ≈ 7.780049; multiplicity 1; Jordan blocks 1 λ ≈ -6.392305; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -5.806615; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 5; Jordan blocks 1, 1, 1, 2 λ ≈ -4; multiplicity 2; Jordan blocks 1, 1 5 2
Sector dimension
13
Spectral radius
$33.0265658$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$33.02657$11
$14.3923$11
$7.780049$11
$-6.392305$11
$-6$11
$-5.806615$11
$-5$51, 1, 1, 2
$-4$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{5} \left(\lambda + 6\right) \left(\lambda^{2} - 8 \lambda - 92\right) \left(\lambda^{3} - 35 \lambda^{2} + 20 \lambda + 1492\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right)^{2} \left(\lambda + 6\right) \left(\lambda^{2} - 8 \lambda - 92\right) \left(\lambda^{3} - 35 \lambda^{2} + 20 \lambda + 1492\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 8 & 64 & 24 & 276 & 144 & 303 & 303 & 656 & 1288 & 724 & 3280 \\ 1 & -1 & 0 & 6 & 8 & 56 & 12 & 38 & 38 & 132 & 234 & 93 & 560 \\ 1 & 0 & -1 & 4 & 16 & 16 & 56 & 36 & 36 & 252 & 174 & 231 & 960 \\ 1 & 2 & 1 & 4 & 0 & 32 & 16 & 40 & 40 & 48 & 150 & 78 & 328 \\ 0 & 1 & 1 & 0 & -2 & 4 & 6 & 3 & 3 & 48 & 32 & 32 & 168 \\ 0 & 1 & 0 & 1 & 1 & 3 & 0 & 4 & 4 & 16 & 32 & 8 & 66 \\ 0 & 0 & 1 & 1 & 2 & 0 & 3 & 4 & 4 & 32 & 16 & 32 & 126 \\ 0 & 0 & 0 & 1 & 0 & 2 & 1 & -2 & 2 & 0 & 8 & 4 & 12 \\ 0 & 0 & 0 & 1 & 0 & 2 & 1 & 2 & -2 & 0 & 8 & 4 & 12 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -\frac{1}{4} & -\frac{1}{4} & 2 & \frac{7}{2} & \frac{15}{4} & 24 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & \frac{3}{2} & \frac{3}{2} & 2 & 0 & \frac{1}{2} & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 4 & 2 & 0 & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$