Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-126-32-d

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w4) + E^(w5) + E^(w1 + w3)
Bundle
O_2(1) + Q_2^v(1) + O_1(1) box U_2^v
ZL62 label
20601.43w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 32
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 126
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 126
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 114
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 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 + 11 0 x0 + 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 28 0 0 x1 + 17 11
$\mathrm{HH}^{ 4 }$ = 32 0 0 0 0 32
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 27t + 70t^{2} + 27t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
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( 2t + 1 \right) \left( 3t + 1 \right) \left( 4t + 1 \right) \left( 16816t^{6} + 12032t^{5} + 3888t^{4} + 808t^{3} + 112t^{2} + 8t - 1 \right) \left( 7501660030272t^{12} + 17028409786176t^{11} + 19770850410992t^{10} + 16077476214032t^{9} + 9147683875352t^{8} + 3425224826372t^{7} + 807614585636t^{6} + 115294113130t^{5} + 9477532082t^{4} + 449225867t^{3} + 18176088t^{2} + 833826t - 10584 \right)D^{6} + 2 \left( 31789274597401595904t^{21} + 121888054907628997632t^{20} + 230205520118183219712t^{19} + 288962129389780277504t^{18} + 267769047874470478720t^{17} + 189776989179930960384t^{16} + 103905385573061113280t^{15} + 44114972776118640448t^{14} + 14567398069418220096t^{13} + 3748704036306761200t^{12} + 751323785930874536t^{11} + 116838567238780656t^{10} + 14062830295561548t^{9} + 1336563884840172t^{8} + 110186194497030t^{7} + 9179337710548t^{6} + 738602963933t^{5} + 43617622866t^{4} + 1524998661t^{3} + 41259540t^{2} + 1672944t - 15876 \right)D^{5} + \left( 529821243290026598400t^{21} + 1932054602550323988480t^{20} + 3496711119934654023168t^{19} + 4245078553351516196608t^{18} + 3811491905511257138304t^{17} + 2597207595182570666432t^{16} + 1348358913507505524544t^{15} + 533797808309431991616t^{14} + 161345970721939856848t^{13} + 37237738027231740656t^{12} + 6541896413468127400t^{11} + 867953332342164976t^{10} + 85735383700265308t^{9} + 6133211029261852t^{8} + 289379853137244t^{7} + 3854099663844t^{6} - 806452858490t^{5} - 81625341493t^{4} - 3596056991t^{3} - 85646694t^{2} - 2796990t + 21168 \right)D^{4} + 2 t \left( 1112624610909055856640t^{20} + 3893283249603917521920t^{19} + 6811833335142182952960t^{18} + 8065777463477036817920t^{17} + 7067816736452221590400t^{16} + 4661587389840982183488t^{15} + 2312849274790364152736t^{14} + 862901794418716377728t^{13} + 242319898507164187552t^{12} + 51218699294577552536t^{11} + 8130269826898625384t^{10} + 966389869704638508t^{9} + 86444858472923560t^{8} + 6036763256273020t^{7} + 361030293831980t^{6} + 19492355128847t^{5} + 757692148305t^{4} + 6255437799t^{3} - 761798076t^{2} - 14707812t + 35868 \right)D^{3} + 4 t \left( 1229185284432861708288t^{20} + 4162505417200814088192t^{19} + 7094900197598352590208t^{18} + 8247541289503461468352t^{17} + 7092136223976227797728t^{16} + 4553399119917567366032t^{15} + 2174920951117207986784t^{14} + 772303913743771143840t^{13} + 204142662827426735300t^{12} + 40193819328116494168t^{11} + 5893709918423565670t^{10} + 646149202484098606t^{9} + 54211349383940813t^{8} + 3716984835271710t^{7} + 224865936536633t^{6} + 11128022014737t^{5} + 295749841976t^{4} - 4854662941t^{3} - 457600766t^{2} - 2831850t + 5292 \right)D^{2} + 8 t ^{2} \left( 667574766545433513984t^{19} + 2205490414505266424832t^{18} + 3687006173302052053632t^{17} + 4229663828691662642624t^{16} + 3585761232180656558368t^{15} + 2253616104180978944976t^{14} + 1044149244084292218848t^{13} + 356392319844254070880t^{12} + 89776911225446251972t^{11} + 16717897422402851904t^{10} + 2307856148610789170t^{9} + 239251310841226086t^{8} + 19452104760008287t^{7} + 1346724937208510t^{6} + 81464156988593t^{5} + 3525894135610t^{4} + 46970485281t^{3} - 3714289479t^{2} - 137757438t - 208656 \right)D + 96 t ^{3} \left( 22706624712429711360t^{18} + 73748658071052948480t^{17} + 121659286971038032512t^{16} + 138326757800882143232t^{15} + 116105482719805814752t^{14} + 71851046053794126624t^{13} + 32553572865889996752t^{12} + 10790752456807330368t^{11} + 2622598095798168568t^{10} + 468538628531365752t^{9} + 61900179979542194t^{8} + 6183576419977886t^{7} + 496822325669221t^{6} + 35019151302960t^{5} + 2103369780312t^{4} + 80326805922t^{3} + 191631069t^{2} - 116414172t - 2786238 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 7)

|λ| = 17.72236 λ ≈ 17.72236; multiplicity 1; Jordan blocks 1 λ ≈ -0.3384991 - 4.624118i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3384991 + 4.624118i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -2.976499 - 2.326802i; multiplicity 1; Jordan blocks 1 λ ≈ -2.976499 + 2.326802i; multiplicity 1; Jordan blocks 1 λ ≈ -3.092362; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$17.7223582$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$17.72236$11
$-0.3384991 - 4.624118\mathrm{i}$11
$-0.3384991 + 4.624118\mathrm{i}$11
$-4$11
$-2.976499 - 2.326802\mathrm{i}$11
$-2.976499 + 2.326802\mathrm{i}$11
$-3.092362$11
$-3$11
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda + 4\right) \left(\lambda^{6} - 8 \lambda^{5} - 112 \lambda^{4} - 808 \lambda^{3} - 3888 \lambda^{2} - 12032 \lambda - 16816\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda + 4\right) \left(\lambda^{6} - 8 \lambda^{5} - 112 \lambda^{4} - 808 \lambda^{3} - 3888 \lambda^{2} - 12032 \lambda - 16816\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 18 & 72 & 18 & 48 & 208 & 168 & 780 \\ 1 & -1 & 3 & 12 & 0 & 12 & 30 & 18 & 120 \\ 1 & 1 & 1 & 12 & 4 & 8 & 42 & 36 & 168 \\ 0 & 1 & 1 & 0 & 0 & 2 & 4 & 0 & 18 \\ 0 & 1 & 0 & 3 & -1 & 0 & 12 & 12 & 36 \\ 0 & 0 & 3 & 3 & 1 & 0 & 12 & 12 & 54 \\ 0 & 0 & 0 & 3 & 0 & 4 & 2 & 3 & 18 \\ 0 & 0 & 0 & 1 & 1 & -2 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{4}{3} & 1 & 0\end{pmatrix}\notag\end{equation}$$