Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-17-10-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + 4w2)
Bundle
O_1(1) box O_2(4)
ZL62 label
140.x303
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 10
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 17
$\chi(X, \mathrm{T}_X)$ = -112

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 17
$\mathrm{c}_1^2\mathrm{c}_2$ = 74
$\mathrm{c}_2^2$ = 276
$\mathrm{c}_1\mathrm{c}_3$ = -94
$\mathrm{c}_4$ = 293
Hodge diamond
1
00
0 2 0
0 0 0 0
0 35 217 35 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 112 0 112 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 180 0 0 170 0 10
$\mathrm{HH}^{ 5 }$ = 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

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

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(745)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(734)

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( 25t + 1 \right) ^{3} \left( 788159051t^{5} + 87274736t^{4} + 2922560t^{3} + 17280t^{2} - 392t + 1 \right) \left( 159651782567133244534t^{9} + 38117403072130158760t^{8} + 3692690172684371156t^{7} + 180798695488337929t^{6} + 4272654323545454t^{5} + 18032474578143t^{4} - 1283358201424t^{3} - 24489608201t^{2} - 122218376t + 20825 \right)D^{6} + 2 \left( 25t + 1 \right) ^{2} \left( 33030636827624646473147949023925t^{15} + 11788830977038718440345127574153t^{14} + 1846120215211441698290439941210t^{13} + 165316733426480309931616895358t^{12} + 9189311532737939745407809575t^{11} + 316592233792619837893945565t^{10} + 5934188476542292094551962t^{9} + 12878230850981020701540t^{8} - 2142051059661837235476t^{7} - 50664289437797835376t^{6} - 503907101515202137t^{5} - 1850985053609013t^{4} + 3650190551460t^{3} + 58835423798t^{2} + 177767289t - 20825 \right)D^{5} + \left( 25t + 1 \right) \left( 13762765344843602697144978759968750t^{16} + 5103812807391114818185697904594500t^{15} + 836851000532720775924957248819838t^{14} + 79313909102602466552057171465157t^{13} + 4743187227478998583348041152812t^{12} + 180978595566705914371126375236t^{11} + 4045858609216490479389369232t^{10} + 28648159252588017338779398t^{9} - 1142044485807767911153287t^{8} - 39905154050224289421486t^{7} - 584711677331235886876t^{6} - 4338371660362029257t^{5} - 13707523771601356t^{4} - 8612073145386t^{3} - 76211898386t^{2} - 231462322t + 20825 \right)D^{4} + 2 t \left( 722545180604289141600111384898359375t^{16} + 282055162680243662225192538816965625t^{15} + 49077761602979587437972601861877375t^{14} + 4990238892289631093729337963565811t^{13} + 325296892044771846064128045647497t^{12} + 13896759185381956134438230715526t^{11} + 370077505886834732961636106026t^{10} + 4549636256482645099844821938t^{9} - 59209785000331907697655019t^{8} - 3722175331009415143462555t^{7} - 75095970143217155196805t^{6} - 819428559302732679929t^{5} - 4825933775666453041t^{4} - 12433464083235716t^{3} - 4290507952444t^{2} - 24762675980t - 788900 \right)D^{3} + 4 t ^{2} \left( 798240390000928956434408768078187500t^{15} + 299096965411397026494702768877757500t^{14} + 49712856563933522718889001783496425t^{13} + 4797188619186946603486357992752522t^{12} + 293850783706918342756481918551793t^{11} + 11582095894972266803371438361764t^{10} + 271178412633518172718415639825t^{9} + 2117697315743944377812924672t^{8} - 82909772813165872168394467t^{7} - 3276680801517985488429472t^{6} - 56324399349094702210942t^{5} - 538224440492869535402t^{4} - 2740283579918123878t^{3} - 5688676093042280t^{2} + 395262659248t - 5717781496 \right)D^{2} + 8 t ^{2} \left( 433527108362573484960066830939015625t^{15} + 157462902321358695090237410109354375t^{14} + 25262461647089275339127795728126700t^{13} + 2339497009548322183515278109593142t^{12} + 136289961769214548461798485625495t^{11} + 5019610844456065047859634197182t^{10} + 104201761413759372184609575181t^{9} + 357543004820394759557353236t^{8} - 46011568641092507772461548t^{7} - 1446657993013485806216829t^{6} - 22110434871094294654557t^{5} - 189449913254898984258t^{4} - 848709200474051640t^{3} - 1451977794114792t^{2} + 134457797096t - 626982440 \right)D + 96 t ^{2} \left( 14745820012332431461226762957109375t^{15} + 5241496237492399714003428787190625t^{14} + 820152697460955689593104983178050t^{13} + 73727050980011912597494625348274t^{12} + 4137686879800352296701456766041t^{11} + 144554445474104843142621631100t^{10} + 2702891774009294935744717165t^{9} - 1666218147022265740045944t^{8} - 1524028430635949108316504t^{7} - 41831151245207888255903t^{6} - 586611377349698571069t^{5} - 4610169324069366758t^{4} - 18561812932134700t^{3} - 26951247315342t^{2} - 4199729030t - 7413700 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ4P4

|λ| = 297.4632 λ ≈ 297.4632; multiplicity 1; Jordan blocks 1 λ ≈ 169.5417; multiplicity 1; Jordan blocks 1 λ ≈ -25.08626; multiplicity 1; Jordan blocks 1 λ ≈ -25; multiplicity 3; Jordan blocks 3 λ ≈ -24.95933 - 0.05612852i; multiplicity 1; Jordan blocks 1 λ ≈ -24.95933 + 0.05612852i; multiplicity 1; Jordan blocks 1 3
Sector dimension
8
Spectral radius
$297.463192$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$297.4632$11
$169.5417$11
$-25.08626$11
$-25$33
$-24.95933 - 0.05612852\mathrm{i}$11
$-24.95933 + 0.05612852\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 25\right)^{3} \left(\lambda^{5} - 392 \lambda^{4} + 17280 \lambda^{3} + 2922560 \lambda^{2} + 87274736 \lambda + 788159051\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 25\right)^{3} \left(\lambda^{5} - 392 \lambda^{4} + 17280 \lambda^{3} + 2922560 \lambda^{2} + 87274736 \lambda + 788159051\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 192 & 4080 & 47088 & 560664 & 8263104 & 24456640 & 384284180 \\ 1 & -1 & 50 & 4096 & 9912 & 566388 & 662361 & 22390864 \\ 1 & 4 & 80 & 832 & 14224 & 164256 & 607452 & 8263104 \\ 0 & 1 & 1 & 79 & 130 & 14016 & 12392 & 548892 \\ 0 & 0 & 1 & 4 & 80 & 832 & 4304 & 47088 \\ 0 & 0 & 0 & 1 & 1 & 79 & 70 & 4032 \\ 0 & 0 & 0 & 0 & 1 & 4 & 0 & 192 \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{5}{4} & 0\end{pmatrix}\notag\end{equation}$$