Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-324-70-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ3P3
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_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2 + w3)
Bundle
O_1(1) box O_2(1) box O_3(1)
ZL62 label
1130.x4000
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 70
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 324
$\chi(X, \mathrm{T}_X)$ = 6

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 324
$\mathrm{c}_1^2\mathrm{c}_2$ = 180
$\mathrm{c}_2^2$ = 92
$\mathrm{c}_1\mathrm{c}_3$ = 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 - 6 0 x0 - 6 x1
$\mathrm{HH}^{ 3 }$ = x1 + 53 0 0 x1 - 31 84
$\mathrm{HH}^{ 4 }$ = 70 0 0 0 0 70
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 65t + 192t^{2} + 65t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(62)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(102)

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( t - 1 \right) \left( 3t + 1 \right) \left( 19t^{2} + 7t + 1 \right) \left( 25t^{2} + 5t - 1 \right) \left( 37t^{3} + 32t^{2} - 2t + 1 \right) \left( 10852743733440t^{13} + 104535214328272t^{12} + 187213361559616t^{11} + 158142799242792t^{10} + 82061402820409t^{9} + 30317418493804t^{8} + 8430053429760t^{7} + 1661240687071t^{6} + 204781425874t^{5} + 14125453584t^{4} + 633093453t^{3} + 29295918t^{2} - 109836t + 2187 \right)D^{6} + \left( 12016429180258104000t^{22} + 128712976189075068000t^{21} + 297394054411305512640t^{20} + 280946198793357091008t^{19} + 80228183048663536165t^{18} - 80486102359551284692t^{17} - 108445151313834759678t^{16} - 69479616531839719237t^{15} - 30685705166211093746t^{14} - 10126996343656829580t^{13} - 2512056594445566678t^{12} - 461508298455847638t^{11} - 63124803776772876t^{10} - 7036893254808412t^{9} - 818543659176110t^{8} - 116325702581652t^{7} - 15912078308537t^{6} - 1545420660598t^{5} - 87822192294t^{4} - 3159151695t^{3} - 117613296t^{2} + 338256t - 4374 \right)D^{5} + \left( 100136909835484200000t^{22} + 1097654452033195980000t^{21} + 2542261964545782194800t^{20} + 2589463636846463309296t^{19} + 1224461686654209401807t^{18} + 9324757576707428568t^{17} - 381256196404464799934t^{16} - 299821838432981169570t^{15} - 144180423174017013062t^{14} - 49558091000071196670t^{13} - 12319025298720462435t^{12} - 2165963949562748476t^{11} - 260975638768623316t^{10} - 20241120675743372t^{9} - 652976842676820t^{8} + 96890053848452t^{7} + 22352649372249t^{6} + 2239860798836t^{5} + 111839438274t^{4} + 2992702734t^{3} + 106447770t^{2} - 354294t + 2187 \right)D^{4} + t \left( 420575021309033640000t^{21} + 4692801545049405480000t^{20} + 10872447258695192234400t^{19} + 11549958348903522457840t^{18} + 6600078436572792302647t^{17} + 1775934179760915734612t^{16} - 299807893716989159466t^{15} - 578645063359459209027t^{14} - 338336902495950427238t^{13} - 126294634298072146780t^{12} - 32285130946178212354t^{11} - 5665182383792762914t^{10} - 682323360606418724t^{9} - 58072669624820930t^{8} - 3772699142129694t^{7} - 192049680261544t^{6} - 6397568304373t^{5} - 121823718542t^{4} - 195502194t^{3} + 55577997t^{2} - 844992t - 972 \right)D^{3} + 2 t ^{2} \left( 464635261636646688000t^{20} + 5254289708128494957000t^{19} + 12166209973283989035560t^{18} + 13230448448475659641044t^{17} + 8278925573537048791518t^{16} + 3121811918294364141300t^{15} + 526468927013422163834t^{14} - 195358090834243000104t^{13} - 200630419582350679404t^{12} - 86875823098246191966t^{11} - 23325587396417614119t^{10} - 4115206947619715234t^{9} - 489384972601446372t^{8} - 41363578772190477t^{7} - 2709131566823778t^{6} - 132075966577454t^{5} - 3428476827447t^{4} - 33072349932t^{3} + 240191274t^{2} - 4644711t - 323352 \right)D^{2} + 4 t ^{2} \left( 252345012785420184000t^{20} + 2881427509480765623000t^{19} + 6666330554317212938040t^{18} + 7343649889311494756228t^{17} + 4820178873486365709823t^{16} + 2073652349982976398228t^{15} + 564211025432123094606t^{14} + 37462163163474138671t^{13} - 53030223818160584686t^{12} - 30245750807431893094t^{11} - 8753664271015820674t^{10} - 1573621789784405194t^{9} - 187365375003292650t^{8} - 15987907745463842t^{7} - 1066202333849458t^{6} - 49704044756794t^{5} - 964097108541t^{4} + 2394265410t^{3} + 211221432t^{2} - 2822445t - 78732 \right)D + 8 t ^{2} \left( 51498982201106160000t^{20} + 591880050474274170000t^{19} + 1368352901996994286800t^{18} + 1517986210443503945088t^{17} + 1022481795925405539430t^{16} + 468408882708440218852t^{15} + 148391858362916168270t^{14} + 24797486575488532716t^{13} - 4133556089222227640t^{12} - 4151238347630604772t^{11} - 1323751092037882245t^{10} - 244434738197916938t^{9} - 29372121150417238t^{8} - 2548261727817451t^{7} - 173411135669382t^{6} - 7831438950480t^{5} - 121037448861t^{4} + 1433845152t^{3} + 36511884t^{2} - 130977t - 8748 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ3P3

|λ| = 8.09017 λ ≈ 8.09017; multiplicity 1; Jordan blocks 1 λ ≈ 1.525473 - 5.734048i; multiplicity 1; Jordan blocks 1 λ ≈ 1.525473 + 5.734048i; multiplicity 1; Jordan blocks 1 λ ≈ -3.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3.09017; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -1.050946; multiplicity 1; Jordan blocks 1 λ ≈ 1; multiplicity 2; Jordan blocks 1, 1 2 2 2
Sector dimension
12
Spectral radius
$8.09016994$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.09017$11
$1.525473 - 5.734048\mathrm{i}$11
$1.525473 + 5.734048\mathrm{i}$11
$-3.5 - 2.598076\mathrm{i}$21, 1
$-3.5 + 2.598076\mathrm{i}$21, 1
$-3.09017$11
$-3$11
$-1.050946$11
$1$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda - 1\right)^{2} \left(\lambda^{2} + 7 \lambda + 19\right)^{2} \left(\lambda^{2} - 5 \lambda - 25\right) \left(\lambda^{3} - 2 \lambda^{2} + 32 \lambda + 37\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda - 1\right) \left(\lambda^{2} + 7 \lambda + 19\right) \left(\lambda^{2} - 5 \lambda - 25\right) \left(\lambda^{3} - 2 \lambda^{2} + 32 \lambda + 37\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 0 & 0 & 0 & 0 & 3 & 0 & 4 & 4 & 10 \\ 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 4 \\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 \\ 3 & 1 & 1 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 3 & 0 & 1 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 1 & 1 & 0 & 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 & 1 & -6 & -2 & -2 & -2 & -4 \\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 4 & 1 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 1 & 2 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 4 & 0\end{pmatrix}\notag\end{equation}$$