Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-26-12-a

·

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

Defining data
Ambient
ℙ1P1 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1 + 3w3)
Bundle
O_1(1) box O_2(3)
ZL62 label
131.x312
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 12
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 26
$\chi(X, \mathrm{T}_X)$ = -81

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 26
$\mathrm{c}_1^2\mathrm{c}_2$ = 80
$\mathrm{c}_2^2$ = 224
$\mathrm{c}_1\mathrm{c}_3$ = -60
$\mathrm{c}_4$ = 186
Hodge diamond
1
00
0 2 0
0 0 0 0
0 20 140 20 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 81 0 81 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 105 0 0 93 0 12
$\mathrm{HH}^{ 5 }$ = 1 0 0 1 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 7t + 10t^{2} + 7t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(739)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(730)

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( 12t + 1 \right) \left( 13t + 1 \right) ^{3} \left( 796t^{2} + 48t - 1 \right) \left( 1724t^{2} + 120t - 1 \right) \left( 3389146538790016t^{9} + 1660775765699520t^{8} + 333714502716988t^{7} + 35075200715992t^{6} + 1997164900093t^{5} + 53279359891t^{4} + 50665536t^{3} - 25881314t^{2} - 322035t + 117 \right)D^{6} + \left( 13t + 1 \right) ^{2} \left( 15236477316390965926846464t^{15} + 10954304970677976473528320t^{14} + 3465197353208335578859264t^{13} + 631962718533421822608704t^{12} + 72916780092867820021248t^{11} + 5463603486388384578992t^{10} + 258627421953986248608t^{9} + 6758040467477945084t^{8} + 35487162522158752t^{7} - 3085864865911719t^{6} - 79314525823753t^{5} - 507497393706t^{4} + 3861870716t^{3} + 113622005t^{2} + 944811t - 234 \right)D^{5} + \left( 13t + 1 \right) \left( 1650618375942354642075033600t^{16} + 1232821754502148891730524160t^{15} + 408333400849728941503009024t^{14} + 78830847859261716292394432t^{13} + 9781561619916736598375024t^{12} + 807754238147557747774400t^{11} + 44014561748388567875020t^{10} + 1470136543459579725804t^{9} + 21573886725485566262t^{8} - 341434452550037899t^{7} - 20879416879486417t^{6} - 330803279926201t^{5} - 1309612113999t^{4} + 8299385783t^{3} - 67745723t^{2} - 606291t + 117 \right)D^{4} + t \left( 90123763326452563457296834560t^{16} + 70775545251081740460870768640t^{15} + 24846144007418054758249662720t^{14} + 5136757782127480701550508224t^{13} + 692124051873701170494439616t^{12} + 63312660538628782510132720t^{11} + 3945816207961768973445536t^{10} + 160777870328535334172740t^{9} + 3642281995639336642172t^{8} + 5351747515847018425t^{7} - 2308654650994757877t^{6} - 67812772361402885t^{5} - 848595557366763t^{4} - 4011873386661t^{3} + 372289569t^{2} - 55937655t - 3537 \right)D^{3} + 2 t ^{2} \left( 99565300436842832009966026752t^{15} + 75257894034739863565967939584t^{14} + 25317729426713846145762435840t^{13} + 4991061197488120031445991232t^{12} + 637268896840686903386267768t^{11} + 54759313569904535912964288t^{10} + 3159567703800301736308606t^{9} + 115497568937691662996854t^{8} + 2079149042447742460431t^{7} - 18929019654684604207t^{6} - 2016078789627131421t^{5} - 47566744517811921t^{4} - 508518856464000t^{3} - 2009966876046t^{2} + 653692506t - 13356486 \right)D^{2} + 4 t ^{2} \left( 54074257995871538074378100736t^{15} + 39705800097711049995804626944t^{14} + 12926391180370403190425318400t^{13} + 2455150168421576262325003776t^{12} + 300325147161640185011631944t^{11} + 24522680933306700953604896t^{10} + 1325716954127042005894962t^{9} + 43914985048069032539746t^{8} + 605193803713862839201t^{7} - 14513440209682908762t^{6} - 868447134733821213t^{5} - 17530624724576578t^{4} - 163253516486118t^{3} - 536579953173t^{2} - 17338911t - 1543464 \right)D + 48 t ^{2} \left( 1839260476050052315455037440t^{15} + 1323719857036449763893084160t^{14} + 421087101815858062182913536t^{13} + 77866771793289981506859648t^{12} + 9229746652091170592066504t^{11} + 725166450490559384488928t^{10} + 37248131834491771016898t^{9} + 1134919288781385599266t^{8} + 11489867025654956821t^{7} - 531508448879225652t^{6} - 24488522456715945t^{5} - 441713779636132t^{4} - 3683252482170t^{3} - 10391560551t^{2} - 8397909t - 18252 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 4)

|λ| = 132.9657 λ ≈ 132.9657; multiplicity 1; Jordan blocks 1 λ ≈ 61.04052; multiplicity 1; Jordan blocks 1 λ ≈ -13.04052; multiplicity 1; Jordan blocks 1 λ ≈ -13; multiplicity 3; Jordan blocks 3 λ ≈ -12.96575; multiplicity 1; Jordan blocks 1 λ ≈ -12; multiplicity 2; Jordan blocks 1, 1 3 2
Sector dimension
9
Spectral radius
$132.965745$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$132.9657$11
$61.04052$11
$-13.04052$11
$-13$33
$-12.96575$11
$-12$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 12\right)^{2} \left(\lambda + 13\right)^{3} \left(\lambda^{2} - 48 \lambda - 796\right) \left(\lambda^{2} - 120 \lambda - 1724\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 12\right) \left(\lambda + 13\right)^{3} \left(\lambda^{2} - 48 \lambda - 796\right) \left(\lambda^{2} - 120 \lambda - 1724\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 72 & 864 & 7236 & 26163 & 26163 & 272172 & 626908 & 6445080 \\ 1 & -1 & 22 & 852 & 946 & 946 & 25875 & 32619 & 536184 \\ 1 & 3 & 30 & 252 & 1266 & 1266 & 10638 & 29421 & 272172 \\ 0 & 1 & 1 & 29 & 25 & 25 & 1224 & 1212 & 24957 \\ 0 & 0 & 1 & 3 & 9 & 21 & 126 & 468 & 3618 \\ 0 & 0 & 1 & 3 & 21 & 9 & 126 & 468 & 3618 \\ 0 & 0 & 0 & 2 & 1 & 1 & 29 & 32 & 840 \\ 0 & 0 & 0 & 0 & 1 & 1 & 3 & 0 & 72 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$