Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-35-14-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ4P4
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + w3) + E^(w1 + 3w3)
Bundle
O_1(1) box O_2(1) + O_1(1) box O_2(3)
ZL62 label
2040.hx312
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 14
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 35
$\chi(X, \mathrm{T}_X)$ = -71

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 35
$\mathrm{c}_1^2\mathrm{c}_2$ = 86
$\mathrm{c}_2^2$ = 204
$\mathrm{c}_1\mathrm{c}_3$ = -46
$\mathrm{c}_4$ = 155
Hodge diamond
1
00
0 2 0
0 0 0 0
0 16 117 16 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
$\mathrm{HH}^{ 2 }$ = x0 + 71 0 71 x0
$\mathrm{HH}^{ 3 }$ = x0 + 2 0 0 x0 + 1 1
$\mathrm{HH}^{ 4 }$ = 85 0 0 71 0 14
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 9t + 15t^{2} + 9t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(729)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(722)

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( 7t + 1 \right) ^{4} \left( 1543623t^{5} + 467362t^{4} + 42187t^{3} + 1395t^{2} + 51t - 1 \right) \left( 374116531215905653t^{10} + 347897113311538035t^{9} + 134302750739171134t^{8} + 27994508336787908t^{7} + 3459781678641240t^{6} + 269314129734207t^{5} + 14559905936639t^{4} + 603862408701t^{3} + 15338109449t^{2} + 54693304t + 20160 \right)D^{6} + \left( 7t + 1 \right) ^{3} \left( 84891747692968219974720393t^{16} + 108128239043473589627219456t^{15} + 60833022436333459981517912t^{14} + 19965219653348615166042961t^{13} + 4260396398394702204000262t^{12} + 626192038551998795787752t^{11} + 66000166839709511884107t^{10} + 5181116680731354587479t^{9} + 314105465036065694202t^{8} + 14902296222284564714t^{7} + 532869062548537544t^{6} + 13213489991291014t^{5} + 210165371481768t^{4} + 2720099361810t^{3} + 60589961020t^{2} + 164583912t + 40320 \right)D^{5} + \left( 7t + 1 \right) ^{2} \left( 4952018615423146165192022925t^{17} + 6611524270736182738521653275t^{16} + 3924161018779203919895264226t^{15} + 1369236943885959583332888445t^{14} + 313419422357691236360543331t^{13} + 49892526517632642457182325t^{12} + 5747419615353196299629472t^{11} + 496513021545323658317275t^{10} + 33287724060936877011829t^{9} + 1759721875431937019897t^{8} + 70863150330954756273t^{7} + 1958133108585371820t^{6} + 30582669440429842t^{5} + 155446621996784t^{4} - 1943977185024t^{3} - 48221848883t^{2} - 113730032t - 20160 \right)D^{4} + t \left( 7t + 1 \right) \left( 145589347293440497256645473995t^{17} + 205859038856449763298381400610t^{16} + 130310998530973315832312218095t^{15} + 48889795224990842065891059763t^{14} + 12144252868987328523141055544t^{13} + 2118467279959731488393065197t^{12} + 269830556731178179063276903t^{11} + 25931405992350486238851301t^{10} + 1939760383576279709634827t^{9} + 115264392206152443067903t^{8} + 5367675630766322416336t^{7} + 183384692333467663778t^{6} + 4073277881741258618t^{5} + 48532312365923698t^{4} + 216266616577806t^{3} + 117319704406t^{2} - 1174289216t + 284544 \right)D^{3} + 2 t ^{2} \left( 1125890952402606512118058332228t^{17} + 1697685135531658886897479666058t^{16} + 1154178499509938958064403550457t^{15} + 468804288274252923271623930481t^{14} + 127189962300311393648503591057t^{13} + 24455710293609545635400937382t^{12} + 3461979501724049123061715300t^{11} + 371840475886659912471320066t^{10} + 31149440728602860876424562t^{9} + 2079332159990419415244341t^{8} + 110632516489285466829935t^{7} + 4512701354478747342152t^{6} + 129624286652602093198t^{5} + 2289443515445638388t^{4} + 20013736028816597t^{3} + 67168894673312t^{2} + 322183084710t + 132930280 \right)D^{2} + 4 t ^{2} \left( 611475258632450088477910990779t^{17} + 900075845265131720992402417409t^{16} + 595375795633879661821419767096t^{15} + 234506655082470292965431313745t^{14} + 61500513964268818732385904756t^{13} + 11400847279542215911514475055t^{12} + 1554048437558825194542346532t^{11} + 160881368008903897571899954t^{10} + 13029921123699154859612276t^{9} + 842348926279330611661189t^{8} + 43114940400579016212256t^{7} + 1657166950246196951828t^{6} + 43218645219415158538t^{5} + 651001027492828410t^{4} + 4271328762679961t^{3} + 9200205314320t^{2} + 76574773568t + 86311680 \right)D + 24 t ^{2} \left( 41596956369554427787612992570t^{17} + 60221180410494449762072336470t^{16} + 39077323281507847053794039404t^{15} + 15058989651360790426413686228t^{14} + 3853886644583665810585726828t^{13} + 695655223953576030505798248t^{12} + 92232676568635162524495753t^{11} + 9294043376626399772835987t^{10} + 734316897758299210624987t^{9} + 46329900441835719190514t^{8} + 2297872823700709056092t^{7} + 84001725761833368758t^{6} + 2017007485008378601t^{5} + 26394770078609564t^{4} + 129583874118336t^{3} + 93844690008t^{2} + 1258849344t + 1854720 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 77.2053 λ ≈ 77.2053; multiplicity 1; Jordan blocks 1 λ ≈ -6.105327 - 19.26308i; multiplicity 1; Jordan blocks 1 λ ≈ -6.105327 + 19.26308i; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 4; Jordan blocks 1, 3 λ ≈ -6.997324 - 0.02812875i; multiplicity 1; Jordan blocks 1 λ ≈ -6.997324 + 0.02812875i; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$77.2053021$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$77.2053$11
$-6.105327 - 19.26308\mathrm{i}$11
$-6.105327 + 19.26308\mathrm{i}$11
$-7$41, 3
$-6.997324 - 0.02812875\mathrm{i}$11
$-6.997324 + 0.02812875\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 7\right)^{4} \left(\lambda^{5} - 51 \lambda^{4} - 1395 \lambda^{3} - 42187 \lambda^{2} - 467362 \lambda - 1543623\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 7\right)^{3} \left(\lambda^{5} - 51 \lambda^{4} - 1395 \lambda^{3} - 42187 \lambda^{2} - 467362 \lambda - 1543623\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 84 & 192 & 4158 & 1458 & 5391 & 116028 & 52668 & 886340 \\ 1 & -1 & 17 & 226 & 66 & 368 & 7242 & 3393 & 61072 \\ 1 & 4 & 9 & 282 & 126 & 338 & 9201 & 4356 & 70224 \\ 0 & 1 & 1 & 8 & 0 & 29 & 290 & 114 & 2952 \\ 0 & 1 & 0 & 17 & -1 & 17 & 354 & 144 & 2604 \\ 0 & 0 & 1 & 4 & 3 & 0 & 234 & 126 & 1944 \\ 0 & 0 & 0 & 1 & 0 & \frac{7}{3} & \frac{13}{3} & 4 & 112 \\ 0 & 0 & 0 & 1 & 1 & -\frac{13}{9} & \frac{209}{9} & \frac{11}{3} & \frac{320}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{4} & \frac{3}{4} & 0\end{pmatrix}\notag\end{equation}$$