Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-464-96-a

·

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

Defining data
Ambient
Fl(1, 2, 4) × ℙ3P3
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3) + E^(w3 + w4)
Bundle
Q_1,1^v ot U_1,2^v ot U_1,3^v + Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1)
ZL62 label
3032.5k
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(0,2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 96
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 464
$\chi(X, \mathrm{T}_X)$ = 16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 464
$\mathrm{c}_1^2\mathrm{c}_2$ = 212
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 0 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 + x1 - 16 0 x0 - 16 x1
$\mathrm{HH}^{ 3 }$ = x1 + 69 0 0 x1 - 77 146
$\mathrm{HH}^{ 4 }$ = 96 0 0 0 0 96
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 91t + 280t^{2} + 91t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(133)
Coates et al. (2020) Obro(4,43)
Coates–Kasprzyk–Prince (2015) CKP(181)

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( 3t + 1 \right) \left( 9t^{2} - 3t + 1 \right) \left( 971t^{6} + 4000t^{5} - 768t^{4} - 50t^{3} + 48t^{2} + t - 1 \right) \left( 633419649180846t^{12} - 744096022660080t^{11} + 2287337812713762t^{10} - 844742150376534t^{9} + 130350191944068t^{8} + 9382932207269t^{7} + 6405854953660t^{6} + 1142389773480t^{5} + 882700722768t^{4} + 33924742700t^{3} - 1817489412t^{2} + 240947920t + 727552 \right)D^{6} + 2 \left( 174366810897029515611t^{21} + 366891618261144137040t^{20} - 119014110507746876193t^{19} + 2187976065036738708771t^{18} - 1215534373644335349150t^{17} + 274322768585402520669t^{16} + 35382713540453367303t^{15} - 14297929315194360141t^{14} + 5079439804811618803t^{13} + 690869368521299605t^{12} + 77545945156021924t^{11} + 14695410257902178t^{10} + 20967427302452232t^{9} - 1378871119398223t^{8} - 338003490723985t^{7} + 57502533807224t^{6} + 4325206084414t^{5} + 2846952335756t^{4} + 125561876358t^{3} - 4456417290t^{2} + 482259616t + 1091328 \right)D^{5} + \left( 2906113514950491926850t^{21} + 4160558611871037021600t^{20} + 1235418955807488504750t^{19} + 30598090304281149847968t^{18} - 17158872122297576920476t^{17} + 4006643105456567002953t^{16} + 193356208514693013780t^{15} + 32081099795315180136t^{14} + 31999446813952181919t^{13} + 23307193469774291664t^{12} - 794308100772901221t^{11} - 220682355542744957t^{10} + 364696364148797602t^{9} - 13498575991534715t^{8} - 4003528951946901t^{7} - 90584697175164t^{6} - 14306679031612t^{5} - 11062086615844t^{4} - 213452395908t^{3} + 7151768716t^{2} - 986572368t - 1455104 \right)D^{4} + 2 t \left( 6102838381396033046385t^{20} + 5512575292334925062400t^{19} + 8045292612103445723445t^{18} + 52774007367916803011229t^{17} - 29930656042495747007298t^{16} + 7034103251197554403734t^{15} + 499705077639162977637t^{14} + 214627130241367465998t^{13} + 42188531017643862704t^{12} + 63755179622874543071t^{11} - 4411481973298016998t^{10} - 306317866425236571t^{9} + 472036309320353824t^{8} - 1812388540628520t^{7} - 3386349374105116t^{6} + 131870102042026t^{5} - 14704905433294t^{4} + 138959011800t^{3} + 26005123160t^{2} + 170344090t - 784 \right)D^{3} + 4 t ^{2} \left( 6742183354685141270292t^{19} + 3364530283982013158340t^{18} + 13581993710138068400688t^{17} + 47477207689509744555390t^{16} - 27327677122579493443734t^{15} + 6390616910024115667095t^{14} + 984780509547247556337t^{13} + 240240942807506667270t^{12} + 50633745517356317429t^{11} + 80485122476712231131t^{10} - 5623791854505859750t^{9} - 548189527091118936t^{8} + 361651955468302095t^{7} + 4407367114596478t^{6} - 2147114262423820t^{5} + 185382634077419t^{4} - 9023497533180t^{3} + 197529963688t^{2} + 4924981312t + 49414144 \right)D^{2} + 8 t ^{2} \left( 3661703028837619827831t^{19} + 742524204019163130540t^{18} + 9268451021852962353027t^{17} + 21166497136579741121547t^{16} - 12401777555961014772612t^{15} + 2871500163553264771710t^{14} + 788105804052284849637t^{13} + 107290392186095249076t^{12} + 33474015307815785175t^{11} + 47699652964699361925t^{10} - 3079006026868764930t^{9} - 390278298017110628t^{8} + 143125108875945968t^{7} + 3094298840172994t^{6} - 709309538502304t^{5} + 103113643299792t^{4} - 2616187118832t^{3} + 78090974128t^{2} + 457081408t + 5820416 \right)D + 480 t ^{3} \left( 24909544413861359373t^{18} + 65721560126878620t^{17} + 71813127513566917449t^{16} + 121893640642352874501t^{15} - 72719608900134336498t^{14} + 16652456695150471944t^{13} + 6731337055988860587t^{12} + 565643033310187878t^{11} + 264629613090030237t^{10} + 343424651408054841t^{9} - 20430766030772288t^{8} - 3091850567810630t^{7} + 761071191048034t^{6} + 20254284190420t^{5} - 3215241181468t^{4} + 666164228872t^{3} - 9749682472t^{2} + 360464988t + 2455488 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 4) × ℙ3P3

|λ| = 6.620917 λ ≈ 6.620917; multiplicity 1; Jordan blocks 1 λ ≈ -5.620152 - 2.200727i; multiplicity 1; Jordan blocks 1 λ ≈ -5.620152 + 2.200727i; multiplicity 1; Jordan blocks 1 λ ≈ 2.925973 - 2.957963i; multiplicity 1; Jordan blocks 1 λ ≈ 2.925973 + 2.957963i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.2325581; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
12
Spectral radius
$6.62091746$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.620917$11
$-5.620152 - 2.200727\mathrm{i}$11
$-5.620152 + 2.200727\mathrm{i}$11
$2.925973 - 2.957963\mathrm{i}$11
$2.925973 + 2.957963\mathrm{i}$11
$-3$21, 1
$1.5 - 2.598076\mathrm{i}$21, 1
$1.5 + 2.598076\mathrm{i}$21, 1
$-0.2325581$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda^{2} - 3 \lambda + 9\right)^{2} \left(\lambda^{6} - \lambda^{5} - 48 \lambda^{4} + 50 \lambda^{3} + 768 \lambda^{2} - 4000 \lambda - 971\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{2} - 3 \lambda + 9\right) \left(\lambda^{6} - \lambda^{5} - 48 \lambda^{4} + 50 \lambda^{3} + 768 \lambda^{2} - 4000 \lambda - 971\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \\ 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & -3 & -3 & 0 \\ 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 2 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 3 & 2 & 0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 3 & 0 & 0 & 0 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 5 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 2 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$