Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-81-24-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ8P8
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11
Bundle
E^(w1 + 2w4) + 2 copies of E^(w3 + w4)
Bundle
O_1(1) box O_2(2) + 2 copies of Q_1 box O_2(1)
ZL62 label
30800.wJwJx321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;9) \\ \det\mathcal{E} &\cong \mathcal{O}(3;8) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 81
$\chi(X, \mathrm{T}_X)$ = -56

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 81
$\mathrm{c}_1^2\mathrm{c}_2$ = 114
$\mathrm{c}_2^2$ = 156
$\mathrm{c}_1\mathrm{c}_3$ = -22
$\mathrm{c}_4$ = 101
Hodge diamond
1
00
0 2 0
0 0 0 0
0 9 77 9 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 56 0 x0 + 56 x2
$\mathrm{HH}^{ 3 }$ = x1 + 12 0 0 x1 12
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 56 0 0 x1 - x2 + 32 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 41t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(567)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(595)

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( 2t + 1 \right) ^{4} \left( 10903t^{5} + 13296t^{4} + 7022t^{3} + 201t^{2} + 7t - 1 \right) \left( 63621945358732t^{11} + 128716433484016t^{10} + 61600833078685t^{9} - 61689817684308t^{8} - 93550033607644t^{7} - 50487817637312t^{6} - 13570564363614t^{5} - 1845247675927t^{4} - 159274063403t^{3} - 16898247925t^{2} - 102148280t - 444528 \right)D^{6} + \left( 2t + 1 \right) ^{3} \left( 29134142950342709832t^{17} + 95366273666527571532t^{16} + 117886551965056012330t^{15} + 36820202694981148995t^{14} - 73003945857427503760t^{13} - 110260606697951722132t^{12} - 76375781702876342981t^{11} - 31926877166864045637t^{10} - 8496526934505044829t^{9} - 1468394127135566955t^{8} - 178221357162499734t^{7} - 17907911647874536t^{6} - 1287287828857983t^{5} - 31456235007167t^{4} - 617304279995t^{3} - 66696442540t^{2} - 301999560t - 889056 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 485569049172378497200t^{18} + 1649930025444145808480t^{17} + 2110705999503038503536t^{16} + 683026320286613553636t^{15} - 1415305815564556979009t^{14} - 2253964458604659980580t^{13} - 1675583094193660019774t^{12} - 771527414839830369963t^{11} - 234505950050295541569t^{10} - 48205131624914199526t^{9} - 7030227224831614625t^{8} - 813855965048801307t^{7} - 75518045717057145t^{6} - 3784301409379090t^{5} - 36775473161397t^{4} + 156194622710t^{3} + 60719425095t^{2} + 258436464t + 444528 \right)D^{4} + t \left( 2t + 1 \right) \left( 4078780013047979376480t^{18} + 14695672422599294906160t^{17} + 20201229079366556713920t^{16} + 7991003180948189891456t^{15} - 13038285211701785125158t^{14} - 23669793249867587784339t^{13} - 19417430984413620730372t^{12} - 9980002980968029264204t^{11} - 3465946675061042495547t^{10} - 836411406630523464295t^{9} - 144607890587056993497t^{8} - 19229859140663278965t^{7} - 2112492330512871038t^{6} - 168930710632792338t^{5} - 6454306371138063t^{4} - 68033028968315t^{3} - 504566096597t^{2} - 2356854528t - 1114064 \right)D^{3} + 2 t ^{2} \left( 9012161552639344908032t^{18} + 34942656614026164777504t^{17} + 52843755876568380740288t^{16} + 27065003659837607486472t^{15} - 29089214434697760845532t^{14} - 65538355468443744034044t^{13} - 60702122583673274998157t^{12} - 35186382499700562038148t^{11} - 13980292248235535910908t^{10} - 3936282749789504072243t^{9} - 803909589577373232720t^{8} - 124256807307340594421t^{7} - 15524699038638088725t^{6} - 1560330870277428189t^{5} - 102974125251909531t^{4} - 2964661979913790t^{3} - 30597732482667t^{2} - 297004717597t - 990587556 \right)D^{2} + 8 t ^{2} \left( 2447268007828787625888t^{18} + 9084002391157080186960t^{17} + 12825266487848483471528t^{16} + 5047599943503348139996t^{15} - 9310135988026375014372t^{14} - 17290875413745790103792t^{13} - 14824398152041828387022t^{12} - 8067010765088969165168t^{11} - 3014457137274467668587t^{10} - 798850575912651833244t^{9} - 154595465924597032467t^{8} - 23071108429814321885t^{7} - 2830857455244896503t^{6} - 271939165931638524t^{5} - 15835657193503459t^{4} - 361172297585047t^{3} - 3934298522902t^{2} - 36700589856t - 95128992 \right)D + 16 t ^{2} \left( 499442450577303597120t^{18} + 1798074245057311375200t^{17} + 2413314202407331498896t^{16} + 734537793501806520264t^{15} - 2062989902705993808880t^{14} - 3439012777062949932624t^{13} - 2798820579914482855366t^{12} - 1455332130019526712296t^{11} - 519496458343731780415t^{10} - 131503694192084173914t^{9} - 24455280179485224552t^{8} - 3563596173435506487t^{7} - 431131450602651915t^{6} - 39658407584604173t^{5} - 2060696006028678t^{4} - 38097782111882t^{3} - 439837414126t^{2} - 3367541072t - 6223392 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 6)

|λ| = 25.95375 λ ≈ 25.95375; multiplicity 1; Jordan blocks 1 λ ≈ -8.499329 - 13.55756i; multiplicity 1; Jordan blocks 1 λ ≈ -8.499329 + 13.55756i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ -0.9775437 - 0.8277093i; multiplicity 1; Jordan blocks 1 λ ≈ -0.9775437 + 0.8277093i; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$25.953746$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$25.95375$11
$-8.499329 - 13.55756\mathrm{i}$11
$-8.499329 + 13.55756\mathrm{i}$11
$-2$41, 3
$-0.9775437 - 0.8277093\mathrm{i}$11
$-0.9775437 + 0.8277093\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda^{5} - 7 \lambda^{4} - 201 \lambda^{3} - 7022 \lambda^{2} - 13296 \lambda - 10903\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{5} - 7 \lambda^{4} - 201 \lambda^{3} - 7022 \lambda^{2} - 13296 \lambda - 10903\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 16 & 288 & 144 & 162 & 960 & 1096 & 2580 \\ 1 & 0 & 7 & -10 & -2 & -6 & 153 & 246 & 504 \\ 1 & 0 & 0 & 44 & 32 & 12 & 288 & 336 & 456 \\ 0 & 1 & 2 & 0 & -1 & 3 & 0 & 12 & 108 \\ 0 & 1 & 0 & 7 & 1 & 6 & 6 & 4 & 54 \\ 0 & 0 & 1 & 0 & 1 & -2 & 30 & 44 & 72 \\ 0 & 0 & 0 & 1 & -\frac{4}{9} & 2 & -\frac{14}{3} & -\frac{28}{9} & \frac{28}{3} \\ 0 & 0 & 0 & 2 & \frac{5}{3} & 0 & 7 & \frac{14}{3} & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{7}{3} & 0\end{pmatrix}\notag\end{equation}$$