Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-193-45-a

·

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

Defining data
Ambient
Fl(2, 3, 6)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w2) + 2 copies of E^(w3) + E^(w5)
Bundle
2 copies of O(1,0) + 2 copies of O(0,1) + wedge^2 Q_1^v ot U_2^v ot wedge^2 U_3^v
ZL62 label
55.42211
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 193
$\chi(X, \mathrm{T}_X)$ = -8

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 193
$\mathrm{c}_1^2\mathrm{c}_2$ = 142
$\mathrm{c}_2^2$ = 106
$\mathrm{c}_1\mathrm{c}_3$ = 42
$\mathrm{c}_4$ = 15
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 9 0 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 + x1 + 8 0 x0 + 8 x1
$\mathrm{HH}^{ 3 }$ = x1 + 39 0 0 x1 + 1 38
$\mathrm{HH}^{ 4 }$ = 45 0 0 0 0 45
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 111t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(385)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( t + 1 \right) ^{2} \left( 19t^{4} + 24t^{3} + 8t^{2} + 11t - 1 \right) \left( 247t^{4} + 27t^{3} + 67t^{2} + 16t + 1 \right) \left( 10721712171495t^{17} + 178714353198852t^{16} + 493306080871227t^{15} + 376253167245640t^{14} - 353291843333951t^{13} - 1063029012867521t^{12} - 1110485280830387t^{11} - 778016836490489t^{10} - 428602303848710t^{9} - 159206760806658t^{8} - 31979497457106t^{7} - 859850923584t^{6} + 461314983958t^{5} + 121004882893t^{4} - 15024375998t^{3} - 359824739t^{2} - 16099856t - 79130 \right)D^{6} + \left( t + 1 \right) \left( 1056656899637346735t^{26} + 20279658365881600062t^{25} + 86873339771388350283t^{24} + 159412455159341482464t^{23} + 112000876381841911421t^{22} - 110622453369062405189t^{21} - 390052829489417368512t^{20} - 560754731418029238657t^{19} - 567462301299178720177t^{18} - 447873716867703592018t^{17} - 283229275514199728225t^{16} - 145748201470408650302t^{15} - 60886465032963156303t^{14} - 19992159025238747945t^{13} - 4816253405428578480t^{12} - 768603137980734125t^{11} - 57097183330159294t^{10} + 7392811720138588t^{9} + 1276174776163496t^{8} + 108824973358474t^{7} - 96639155673410t^{6} - 3988549974474t^{5} + 390261578186t^{4} - 83188218814t^{3} - 1329893156t^{2} - 47033488t - 158260 \right)D^{5} + \left( 8805474163644556125t^{27} + 178868869248719269815t^{26} + 850931206427175596097t^{25} + 1782844208307002669827t^{24} + 1550620217695999614581t^{23} - 1034054903707124543505t^{22} - 4966613339471143438214t^{21} - 7819887517260642489942t^{20} - 8275207448259446309787t^{19} - 6755006476974408390743t^{18} - 4423849451776099954430t^{17} - 2339608349242097263117t^{16} - 993384942744196972339t^{15} - 328848401304423581003t^{14} - 77830039047213836682t^{13} - 9907813518057693081t^{12} + 576262199619702281t^{11} + 532303372566666529t^{10} + 112373991477984999t^{9} + 13129848823105624t^{8} + 571246760754200t^{7} - 315182300890682t^{6} - 22883225169088t^{5} - 2543304339249t^{4} + 43543055747t^{3} + 1291082573t^{2} + 38803582t + 79130 \right)D^{4} + t \left( 36982991487307135725t^{26} + 754767849563623486695t^{25} + 3425202123622843032795t^{24} + 6592256325241651646361t^{23} + 4342150476096638768499t^{22} - 7125110270181983191302t^{21} - 22287300609179247717481t^{20} - 31444995988351656923191t^{19} - 30970470273800046518730t^{18} - 23781339785073364982553t^{17} - 14586524876802439326239t^{16} - 7090665511102121881725t^{15} - 2684132458889598700813t^{14} - 760036880213643655090t^{13} - 141035384964511748759t^{12} - 7459171443927382809t^{11} + 4054476558775422671t^{10} + 1226117687691237550t^{9} + 128280327603021222t^{8} + 1858211954249002t^{7} - 1540751176837638t^{6} - 394288624399924t^{5} - 10120881454978t^{4} - 1237305082118t^{3} - 28739000930t^{2} - 178535136t - 36080 \right)D^{3} + 2 t ^{2} \left( 40857400119310740420t^{25} + 836812820816112391212t^{24} + 3652092183248319359670t^{23} + 6524984078219702804414t^{22} + 3076969780786025167187t^{21} - 9841926933348819331969t^{20} - 25228145384550988500736t^{19} - 33152081399639071766818t^{18} - 31189710058897098480002t^{17} - 23050877961833669626625t^{16} - 13491533601176028791616t^{15} - 6106641854896683802143t^{14} - 2072488262923619324713t^{13} - 500283943776257994892t^{12} - 70452851554915266999t^{11} + 1788875184864704614t^{10} + 3281889099683139824t^{9} + 684774870831269749t^{8} + 29178681110608438t^{7} - 5651279365254663t^{6} - 1035355514417322t^{5} - 76446955058955t^{4} - 2316986871945t^{3} - 608905792712t^{2} - 11105964334t - 72669132 \right)D^{2} + 4 t ^{2} \left( 22189794892384281435t^{25} + 455659595082835214097t^{24} + 1929228419292883625295t^{23} + 3242418390570989158191t^{22} + 1018736037816904222138t^{21} - 6030229879323130041927t^{20} - 13798108283934330096737t^{19} - 17295089465137696828497t^{18} - 15801909249419150681612t^{17} - 11396348613319363322174t^{16} - 6443325281620814542259t^{15} - 2748427544715141909706t^{14} - 845125418753601079076t^{13} - 174017559452956044024t^{12} - 17305307482368481592t^{11} + 2337739267010770337t^{10} + 1193480486379337431t^{9} + 188256499344020743t^{8} - 4941839018868416t^{7} - 3417716712583385t^{6} - 284446561185916t^{5} + 13354922122093t^{4} - 615547036939t^{3} - 165500463812t^{2} - 2416596632t - 15192960 \right)D + 8 t ^{2} \left( 4528529569874343150t^{25} + 93154954410179846730t^{24} + 386097426487309991166t^{23} + 620223510174247407528t^{22} + 120686600209505724540t^{21} - 1317042596711939090734t^{20} - 2817957710898401586427t^{19} - 3426393368518152677669t^{18} - 3074094388038417801282t^{17} - 2182980616125290146604t^{16} - 1204148650030720126703t^{15} - 490717404432053328434t^{14} - 139097598205871888092t^{13} - 24698984750205726365t^{12} - 1508887826874095956t^{11} + 525948371645020448t^{10} + 170070901931722583t^{9} + 20792125698518187t^{8} - 2363186247943124t^{7} - 611192802331943t^{6} - 26277401550389t^{5} + 5205890975534t^{4} - 117557433872t^{3} - 19306434102t^{2} - 233959530t - 1266080 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 6)

|λ| = 11.85683 λ ≈ 11.85683; multiplicity 1; Jordan blocks 1 λ ≈ -8.215941 - 1.75962i; multiplicity 1; Jordan blocks 1 λ ≈ -8.215941 + 1.75962i; multiplicity 1; Jordan blocks 1 λ ≈ 0.215941 - 1.857971i; multiplicity 1; Jordan blocks 1 λ ≈ 0.215941 + 1.857971i; multiplicity 1; Jordan blocks 1 λ ≈ -0.04508497 - 1.445039i; multiplicity 1; Jordan blocks 1 λ ≈ -0.04508497 + 1.445039i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ -0.766661; multiplicity 1; Jordan blocks 1 2
Sector dimension
10
Spectral radius
$11.8568309$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.85683$11
$-8.215941 - 1.75962\mathrm{i}$11
$-8.215941 + 1.75962\mathrm{i}$11
$0.215941 - 1.857971\mathrm{i}$11
$0.215941 + 1.857971\mathrm{i}$11
$-0.04508497 - 1.445039\mathrm{i}$11
$-0.04508497 + 1.445039\mathrm{i}$11
$-1$22
$-0.766661$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{4} + 16 \lambda^{3} + 67 \lambda^{2} + 27 \lambda + 247\right) \left(\lambda^{4} - 11 \lambda^{3} - 8 \lambda^{2} - 24 \lambda - 19\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{4} + 16 \lambda^{3} + 67 \lambda^{2} + 27 \lambda + 247\right) \left(\lambda^{4} - 11 \lambda^{3} - 8 \lambda^{2} - 24 \lambda - 19\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 9 & 3 & 15 & 6 & 40 & 40 & 45 \\ 1 & -1 & 1 & 6 & 2 & 10 & 4 & 0 & 3 & 24 \\ 1 & 2 & -1 & 0 & 4 & 2 & 6 & 15 & 9 & 0 \\ 0 & 1 & 0 & -1 & 1 & 0 & 2 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & -1 & 1 & -2 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & -1 & 0 & 10 & 8 & 0 \\ 0 & 0 & 1 & 1 & 0 & 2 & 0 & 0 & 2 & 9 \\ 0 & 0 & 0 & 2 & -\frac{6}{5} & \frac{13}{5} & -\frac{11}{5} & -4 & -\frac{7}{5} & \frac{24}{5} \\ 0 & 0 & 0 & 0 & 3 & 1 & 5 & 5 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{10}{3} & 3 & 0\end{pmatrix}\notag\end{equation}$$