Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-63-20-a

·

This is a Fano fourfold of K3 type, appearing as case C-10 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 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_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(3w3) + E^(w2 + w3) + E^(w1 + w3)
Bundle
O_3(3) + O_2(1) box O_3(1) + O_1(1) box O_3(1)
ZL62 label
1150.Gw6w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 20
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 63
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 63
$\mathrm{c}_1^2\mathrm{c}_2$ = 102
$\mathrm{c}_2^2$ = 141
$\mathrm{c}_1\mathrm{c}_3$ = 0
$\mathrm{c}_4$ = 48
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 38 1 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 36 0 x0 + 36 x2
$\mathrm{HH}^{ 3 }$ = x1 + 4 0 0 x1 4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + x3 + 13 0 0 x1 - x2 - 7 x3 20
$\mathrm{HH}^{ 5 }$ = x3 - 2 0 0 x3 - 2 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 15t + 31t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(711)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(708)

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( 8t + 1 \right) \left( 20t - 1 \right) \left( 6t + 1 \right) ^{2} \left( 7t + 1 \right) ^{2} \left( 459t^{3} - 297t^{2} - 36t + 1 \right) \left( 174701351044850688t^{13} + 210036807597072384t^{12} + 113417802265660416t^{11} + 35953764283991808t^{10} + 7352334112283136t^{9} + 1009130747356416t^{8} + 95197935993696t^{7} + 6455881345696t^{6} + 346170060438t^{5} + 14842705572t^{4} + 296527525t^{3} - 10780785t^{2} - 518079t + 693 \right)D^{6} + \left( 6t + 1 \right) \left( 7t + 1 \right) \left( 11316119288687242052567040t^{20} + 11180635714529675516583936t^{19} + 2454830492533168955774976t^{18} - 1896170403500121535423488t^{17} - 1643519808167528275992576t^{16} - 630591519694017471768576t^{15} - 152338546534521021253632t^{14} - 25431508444168064335104t^{13} - 3063182655253061509056t^{12} - 275812806990651274224t^{11} - 19389178386301826844t^{10} - 1085888064082233358t^{9} - 42124391628238050t^{8} - 300151589112261t^{7} + 74841802648894t^{6} + 3623637155094t^{5} + 37488888657t^{4} - 687677416t^{3} - 30295011t^{2} - 1541763t + 1386 \right)D^{5} + \left( 3960641751040534718398464000t^{22} + 5292175659252828454558679040t^{21} + 2669683566992112265363845120t^{20} + 319661254309568268293443584t^{19} - 328392623227967751100511232t^{18} - 220073228912775314471003136t^{17} - 73914126224115824821301760t^{16} - 16510299053250827501298432t^{15} - 2666118316063242830606208t^{14} - 325167969808312464214368t^{13} - 31074384508877557195224t^{12} - 2413294492733759382648t^{11} - 154624912354521878130t^{10} - 7734626250513217438t^{9} - 247721394586773953t^{8} - 1671957438200940t^{7} + 192246988565001t^{6} + 3762005383615t^{5} - 179937628824t^{4} - 6041039294t^{3} + 7363443t^{2} + 956661t - 693 \right)D^{4} + t \left( 16634695354370245817273548800t^{21} + 22732170548408749153742929920t^{20} + 12677757850217539966599475200t^{19} + 3018083463939715231715168256t^{18} - 288787275385410585092207616t^{17} - 458778385510973776625366016t^{16} - 170614076220548695545246720t^{15} - 38647904054946065396711424t^{14} - 6162008046893934878185344t^{13} - 738238093060840324725600t^{12} - 69926905379215207257624t^{11} - 5447812270091405836440t^{10} - 346353635723551603874t^{9} - 16101232313752212854t^{8} - 376523430537614481t^{7} + 8143377704277071t^{6} + 873437007595442t^{5} + 21269951552763t^{4} + 67138767715t^{3} - 1551992435t^{2} + 35111778t + 6039 \right)D^{3} + 2 t ^{2} \left( 18377377724828081093368872960t^{20} + 25540508979049519836115402752t^{19} + 15119464281424379482654679040t^{18} + 4526391072736856122075717632t^{17} + 417809535417846414827661312t^{16} - 206267277578473106972914176t^{15} - 102039542539151633554961664t^{14} - 24523233548295657384596736t^{13} - 3953883064068533185891200t^{12} - 474867360176222842364736t^{11} - 45555523011582253130784t^{10} - 3631659648818620693272t^{9} - 232471851969874259060t^{8} - 10234493452277278544t^{7} - 173779905422150921t^{6} + 9677818992881513t^{5} + 673100461969765t^{4} + 14548959751696t^{3} + 68162879053t^{2} - 149450727t + 9655527 \right)D^{2} + 24 t ^{2} \left( 1663469535437024581727354880t^{20} + 2340172158189891724317007872t^{19} + 1436699668085999588010240000t^{18} + 478913161588765613730035712t^{17} + 78852471834995141822885376t^{16} - 2534172967308979953402624t^{15} - 4754772089780448199552128t^{14} - 1310100514183759444286400t^{13} - 220003991346918028765536t^{12} - 27050382115944371049888t^{11} - 2674534320515183249268t^{10} - 219963176043084053580t^{9} - 14119685294460521757t^{8} - 580229099707743261t^{7} - 5590732002714087t^{6} + 818264134158118t^{5} + 44338951472935t^{4} + 847334951815t^{3} + 4055217799t^{2} + 15090606t + 233310 \right)D + 144 t ^{2} \left( 113161192886872420525670400t^{20} + 160496747463821921167196160t^{19} + 100704979408515169121464320t^{18} + 35488902810282171432726528t^{17} + 7040007649847719729465344t^{16} + 469698897234214552932864t^{15} - 149554948029254378820864t^{14} - 54711755720106347597760t^{13} - 9883291008171784885920t^{12} - 1266635739015824748336t^{11} - 130565189895359762880t^{10} - 11102286965641257860t^{9} - 711907547741454620t^{8} - 27104213323370418t^{7} - 44276935493267t^{6} + 51424695640529t^{5} + 2364339253672t^{4} + 40727631544t^{3} + 195628537t^{2} + 1439427t + 6006 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ5P5

|λ| = 42.70327 λ ≈ 42.70327; multiplicity 1; Jordan blocks 1 λ ≈ 20; multiplicity 2; Jordan blocks 1, 1 λ ≈ -8.040134; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 4; Jordan blocks 2, 2 λ ≈ -6; multiplicity 2; Jordan blocks 2 λ ≈ 1.336867; multiplicity 1; Jordan blocks 1 2 4 2
Sector dimension
12
Spectral radius
$42.7032672$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$42.70327$11
$20$21, 1
$-8.040134$11
$-8$11
$-7$42, 2
$-6$22
$1.336867$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda + 7\right)^{4} \left(\lambda + 8\right) \left(\lambda - 20\right)^{2} \left(\lambda^{3} - 36 \lambda^{2} - 297 \lambda + 459\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda + 7\right)^{2} \left(\lambda + 8\right) \left(\lambda - 20\right) \left(\lambda^{3} - 36 \lambda^{2} - 297 \lambda + 459\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 132 & 18 & 342 & 342 & 1530 & 864 & 4104 & 4104 & 11700 \\ 1 & -1 & 0 & 6 & 12 & 120 & 12 & 156 & 324 & 1188 & 432 & 3240 \\ 1 & 0 & -1 & 6 & 12 & 12 & 120 & 156 & 324 & 432 & 1188 & 3240 \\ 1 & 1 & 1 & 9 & 0 & 30 & 30 & 168 & 45 & 369 & 369 & 864 \\ 0 & 1 & 1 & 0 & -2 & 6 & 6 & 6 & 108 & 144 & 144 & 864 \\ 0 & 1 & 0 & 1 & 1 & 8 & 0 & 15 & 30 & 138 & 30 & 324 \\ 0 & 0 & 1 & 1 & 1 & 0 & 8 & 15 & 30 & 30 & 138 & 324 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 12 & 12 & 18 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -1 & 7 & 14 & 14 & 108 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 1 & 0 & 1 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$