Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-83-24-a

·

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

Defining data
Ambient
ℙ4P4 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 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_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(w1 + w5) + E^(w4 + w5) + E^(2w1 + w5)
Bundle
O_1(1) box O_2(1) + Q_1 box O_2(1) + O_1(2) box O_2(1)
ZL62 label
40600.wOw3x337
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;7) \\ \det\mathcal{E} &\cong \mathcal{O}(4;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 83
$\chi(X, \mathrm{T}_X)$ = -45

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 83
$\mathrm{c}_1^2\mathrm{c}_2$ = 110
$\mathrm{c}_2^2$ = 150
$\mathrm{c}_1\mathrm{c}_3$ = -10
$\mathrm{c}_4$ = 77
Hodge diamond
1
00
0 2 0
0 0 0 0
0 6 59 6 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 + 45 0 x0 + 45 x1
$\mathrm{HH}^{ 3 }$ = x1 + 12 0 0 x1 + 1 11
$\mathrm{HH}^{ 4 }$ = 43 0 0 19 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 43t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(635)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(645)

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) ^{4} \left( 10037t^{5} + 16762t^{4} + 7225t^{3} + 484t^{2} - 8t - 1 \right) \left( 210542598834t^{10} + 839886505626t^{9} + 1446999826588t^{8} + 1387695815775t^{7} + 789488102397t^{6} + 264429177553t^{5} + 48532477675t^{4} + 3206047337t^{3} - 68993547t^{2} + 3093229t + 2573 \right)D^{6} + 2 \left( t + 1 \right) ^{3} \left( 22188768677217009t^{16} + 131179691538709032t^{15} + 348604847686969772t^{14} + 548307392339122344t^{13} + 564294125293906596t^{12} + 396302204602805776t^{11} + 192353733542921236t^{10} + 63904533274897913t^{9} + 13950764250202367t^{8} + 1806107314435553t^{7} + 109144331274255t^{6} + 1623511022073t^{5} + 196957741879t^{4} + 11999955849t^{3} - 162244539t^{2} + 4605108t + 2573 \right)D^{5} + \left( t + 1 \right) ^{2} \left( 369812811286950150t^{17} + 2335085482611347190t^{16} + 6684661215913962140t^{15} + 11440847925153671429t^{14} + 12973208236214694693t^{13} + 10199547455844686599t^{12} + 5657471675843284187t^{11} + 2209736807877309216t^{10} + 593491157598361601t^{9} + 103192353205307039t^{8} + 10161405538035422t^{7} + 391782820926241t^{6} - 4260828390057t^{5} + 310770657409t^{4} - 9893254885t^{3} + 205711306t^{2} - 6779395t - 2573 \right)D^{4} + 2 t \left( t + 1 \right) \left( 776606903702595315t^{17} + 5315544807557181150t^{16} + 16603236209645432075t^{15} + 31251160298698463110t^{14} + 39365812968526924380t^{13} + 34830331319158627977t^{12} + 22108960944290217366t^{11} + 10104872409179018320t^{10} + 3282573321678841705t^{9} + 730958541505549430t^{8} + 103409517636317961t^{7} + 7955816829217186t^{6} + 224906298863048t^{5} + 2355314437695t^{4} + 842207318638t^{3} + 9162497600t^{2} + 7108840t + 1860 \right)D^{3} + 4 t ^{2} \left( 857965722185724348t^{17} + 6422083625252749812t^{16} + 22048732138381380251t^{15} + 45908437371546303696t^{14} + 64508145394216119487t^{13} + 64368006134605644234t^{12} + 46732485400609573234t^{11} + 24883841267873435586t^{10} + 9663390997786880530t^{9} + 2678206900825162907t^{8} + 505874572241987874t^{7} + 59638709255463859t^{6} + 3709277211634552t^{5} + 94181076130201t^{4} + 5151419992609t^{3} + 461341592679t^{2} + 2647967415t + 4027326 \right)D^{2} + 8 t ^{2} \left( 465964142221557189t^{17} + 3365155655459876919t^{16} + 11145213959025407240t^{15} + 22368263509697984070t^{14} + 30245942553571287884t^{13} + 28965121952075443653t^{12} + 20107575307758734339t^{11} + 10187018629688602711t^{10} + 3738400783165922206t^{9} + 969123031245427528t^{8} + 168472800745129703t^{7} + 17829722761415149t^{6} + 967721191270907t^{5} + 26432509334808t^{4} + 2442325407190t^{3} + 121623102390t^{2} + 409411974t + 895404 \right)D + 96 t ^{2} \left( 15849120483726435t^{17} + 111641490989441985t^{16} + 360484189710458690t^{15} + 704693162385833046t^{14} + 926579358090956120t^{13} + 860692700658935761t^{12} + 577571274123593225t^{11} + 281583721270917161t^{10} + 98820105350542042t^{9} + 24268422816861716t^{8} + 3938324633616667t^{7} + 380965297953831t^{6} + 18676255026745t^{5} + 592158078430t^{4} + 64960324338t^{3} + 2121931690t^{2} + 4561650t + 10292 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ6P6

|λ| = 24.65751 λ ≈ 24.65751; multiplicity 1; Jordan blocks 1 λ ≈ -14.97633 - 3.773393i; multiplicity 1; Jordan blocks 1 λ ≈ -14.97633 + 3.773393i; multiplicity 1; Jordan blocks 1 λ ≈ -1.702472; multiplicity 1; Jordan blocks 1 λ ≈ -1.002383; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
9
Spectral radius
$24.657513$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$24.65751$11
$-14.97633 - 3.773393\mathrm{i}$11
$-14.97633 + 3.773393\mathrm{i}$11
$-1.702472$11
$-1.002383$11
$-1$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{4} \left(\lambda^{5} + 8 \lambda^{4} - 484 \lambda^{3} - 7225 \lambda^{2} - 16762 \lambda - 10037\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} + 8 \lambda^{4} - 484 \lambda^{3} - 7225 \lambda^{2} - 16762 \lambda - 10037\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 24 & 144 & 72 & 180 & 1320 & 1176 & 3630 \\ 1 & -2 & 4 & 32 & 44 & 24 & 324 & 222 & 432 \\ 1 & 2 & -1 & 52 & 28 & 60 & 108 & 102 & 888 \\ 0 & 1 & 1 & -3 & -3 & 0 & 36 & 36 & 54 \\ 0 & 1 & 0 & 4 & -2 & 6 & 0 & 8 & 108 \\ 0 & 0 & 1 & 2 & 5 & -1 & 48 & 32 & 18 \\ 0 & 0 & 0 & 1 & -\frac{5}{6} & 2 & -6 & -\frac{4}{3} & 24 \\ 0 & 0 & 0 & 1 & \frac{5}{2} & 0 & 9 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$