Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-51-18-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ7P7
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(3w3) + 2 copies of E^(w2 + w3)
Bundle
O_2(3) + 2 copies of Q_1 box O_2(1)
ZL62 label
20700.Kw8w8
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;8) \\ \det\mathcal{E} &\cong \mathcal{O}(2;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 51
$\chi(X, \mathrm{T}_X)$ = -70

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 51
$\mathrm{c}_1^2\mathrm{c}_2$ = 102
$\mathrm{c}_2^2$ = 180
$\mathrm{c}_1\mathrm{c}_3$ = -42
$\mathrm{c}_4$ = 135
Hodge diamond
1
00
0 2 0
0 0 0 0
0 13 103 13 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 + x2 + 70 0 x0 + 70 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 70 0 0 x1 - x2 + 52 0 18
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 23t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(705)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(702)

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( 5t + 1 \right) \left( 6t + 1 \right) ^{4} \left( 119475t^{4} + 18495t^{3} - 729t^{2} + 57t - 1 \right) \left( 6991223932372500t^{11} + 9237395296699200t^{10} + 5243910253576335t^{9} + 1682707188881556t^{8} + 335077264500669t^{7} + 42388579451131t^{6} + 3347000119860t^{5} + 161178118935t^{4} + 5196335810t^{3} + 128272899t^{2} - 74544t + 136 \right)D^{6} + \left( 6t + 1 \right) ^{3} \left( 526224181971728795625000t^{17} + 908778197876045362762500t^{16} + 700434499826576815841250t^{15} + 318730747876774026643125t^{14} + 95221126404650356146000t^{13} + 19593875205271540920735t^{12} + 2829127279667832053802t^{11} + 287046932104061240976t^{10} + 20377010470976318331t^{9} + 1021071885160180920t^{8} + 37461484107976097t^{7} + 1015943639830148t^{6} + 21820771033692t^{5} + 743245160662t^{4} + 18114876790t^{3} + 514451820t^{2} - 219280t + 272 \right)D^{5} + \left( 6t + 1 \right) ^{2} \left( 26311209098586439781250000t^{18} + 47387184315856835994000000t^{17} + 38378646886157217249480000t^{16} + 18526974843106910276194500t^{15} + 5942095705326101570033625t^{14} + 1332637413277940513948070t^{13} + 213877706675595805892043t^{12} + 24744600568500458759505t^{11} + 2064972167089084705392t^{10} + 124923028332259022064t^{9} + 5571657967503871752t^{8} + 181046285574699370t^{7} + 3431805221220989t^{6} - 4279721744214t^{5} - 1403487320791t^{4} - 27482764226t^{3} - 402418329t^{2} + 161344t - 136 \right)D^{4} + 3 t \left( 6t + 1 \right) \left( 221014156428126094162500000t^{18} + 418804322168333502132750000t^{17} + 359373106750108276279800000t^{16} + 185349493731595764978738000t^{15} + 64153708639125965276948550t^{14} + 15720723836793410250418965t^{13} + 2800114644882903744234348t^{12} + 366568350399608167562247t^{11} + 35392742342536347921690t^{10} + 2529126746388446089836t^{9} + 135214782929943955633t^{8} + 5463918839141310000t^{7} + 159264764451121165t^{6} + 2693048523372168t^{5} + 11372490633592t^{4} - 157417821196t^{3} - 3021548158t^{2} + 1336368t + 448 \right)D^{3} + 6 t ^{2} \left( 1465008122609292967020000000t^{18} + 2938674953348919200331900000t^{17} + 2686678793690972919277680000t^{16} + 1487286451514629465796799000t^{15} + 557329224351287607333700500t^{14} + 149415533129065954694813460t^{13} + 29494394247671360504020683t^{12} + 4346892328990170199165146t^{11} + 481017005265786568654734t^{10} + 40081410080161319096754t^{9} + 2528642916620123995080t^{8} + 121700012507047106515t^{7} + 4407635703158361686t^{6} + 108820597770103590t^{5} + 1329958747089332t^{4} - 934477568754t^{3} - 80968189578t^{2} - 320484008t + 348976 \right)D^{2} + 72 t ^{2} \left( 132608493856875656497500000t^{18} + 260590141698336057130650000t^{17} + 232899601762753466571495000t^{16} + 125822225642239296728959500t^{15} + 45934978582822706352740700t^{14} + 11972372577763491453876480t^{13} + 2291410720933871397114096t^{12} + 326406796766220271680558t^{11} + 34809142597412727651069t^{10} + 2791196828475536373872t^{9} + 169613604367936339765t^{8} + 7866331339989547708t^{7} + 270946563026722414t^{6} + 6033341527774643t^{5} + 53720429035230t^{4} - 399245695807t^{3} - 5170706490t^{2} + 2338536t + 9520 \right)D + 48 t ^{2} \left( 81188873789923871325000000t^{18} + 157306990442921203450500000t^{17} + 138384808892229386291490000t^{16} + 73486451829102022884339000t^{15} + 26333153441811729421066800t^{14} + 6724597051969274708699400t^{13} + 1258068238688908759792122t^{12} + 174701385297220829838822t^{11} + 18116139264818401388169t^{10} + 1410665707014006713022t^{9} + 83292707884585881948t^{8} + 3751505102477195376t^{7} + 123891929817870514t^{6} + 2524842026839863t^{5} + 16311219320340t^{4} - 242574428110t^{3} - 2443751034t^{2} + 254280t + 2992 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ7P7

|λ| = 50.73147 λ ≈ 50.73147; multiplicity 1; Jordan blocks 1 λ ≈ 5.651002 - 20.87925i; multiplicity 1; Jordan blocks 1 λ ≈ 5.651002 + 20.87925i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 4; Jordan blocks 1, 3 λ ≈ -5.033477; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$50.7314732$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$50.73147$11
$5.651002 - 20.87925\mathrm{i}$11
$5.651002 + 20.87925\mathrm{i}$11
$-6$41, 3
$-5.033477$11
$-5$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda + 6\right)^{4} \left(\lambda^{4} - 57 \lambda^{3} + 729 \lambda^{2} - 18495 \lambda - 119475\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda + 6\right)^{3} \left(\lambda^{4} - 57 \lambda^{3} + 729 \lambda^{2} - 18495 \lambda - 119475\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 120 & 864 & 432 & 2106 & 28152 & 12168 & 217080 \\ 1 & 0 & 12 & 84 & -12 & 252 & 756 & 108 & 7632 \\ 1 & 0 & 9 & 54 & 42 & 174 & 2754 & 1323 & 24336 \\ 0 & 1 & 1 & 9 & -2 & 30 & 48 & -30 & 0 \\ 0 & 1 & 0 & 12 & 1 & 18 & 288 & 108 & 1620 \\ 0 & 0 & 1 & 0 & 1 & 0 & 72 & 42 & 864 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 24 \\ 0 & 0 & 0 & 1 & 1 & -1 & 30 & 9 & 192 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$