Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-304-66-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ4P4
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_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w3 + w4) + E^(w2 + w4) + E^(w1 + w4)
Bundle
O_3(1) box O_4(1) + O_2(1) box O_4(1) + O_1(1) box O_4(1)
ZL62 label
11140.wBw7w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 66
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 304
$\chi(X, \mathrm{T}_X)$ = 6

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 304
$\mathrm{c}_1^2\mathrm{c}_2$ = 172
$\mathrm{c}_2^2$ = 97
$\mathrm{c}_1\mathrm{c}_3$ = 62
$\mathrm{c}_4$ = 17
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 7 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + 24 0 x0 - 6 30
$\mathrm{HH}^{ 3 }$ = 74 0 0 0 74
$\mathrm{HH}^{ 4 }$ = 66 0 0 0 0 66
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 61t + 180t^{2} + 61t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(107)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(153)

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( 53t^{4} + 45t^{3} - t^{2} - 5t - 1 \right) \left( 3077t^{5} + 1859t^{4} + 300t^{3} + 8t^{2} - 1 \right) \left( 2345368151422792t^{13} + 8149827031300300t^{12} + 11666729533119287t^{11} + 9312130072981790t^{10} + 4718696680228063t^{9} + 1596309836805827t^{8} + 362667775843295t^{7} + 55312516420183t^{6} + 6280898637731t^{5} + 653497062886t^{4} + 50789538396t^{3} + 1552051052t^{2} + 7350522t + 448400 \right)D^{6} + \left( 8032184653545787185192t^{22} + 38688972038026488402888t^{21} + 81467886247887566396969t^{20} + 99918491943310306091776t^{19} + 79845036795432307598445t^{18} + 43591200372166795232518t^{17} + 16229513088972998917903t^{16} + 3767388269273007933540t^{15} + 289360532966081934644t^{14} - 148183495848674279115t^{13} - 69997540570426113844t^{12} - 17142922750903246466t^{11} - 3108091531971521726t^{10} - 487962669876221126t^{9} - 68932073225378200t^{8} - 8139636591295836t^{7} - 748007795751821t^{6} - 56774335461973t^{5} - 4265326449482t^{4} - 255931300250t^{3} - 6190284398t^{2} - 17567566t - 896800 \right)D^{5} + \left( 66934872112881559876600t^{22} + 316446420225710889608360t^{21} + 652946782046336824085687t^{20} + 787147739857779929199854t^{19} + 623793955298871843493807t^{18} + 343680525146851928146663t^{17} + 134012623279247661377975t^{16} + 36233971959442140449491t^{15} + 6178639440887914358670t^{14} + 396235223497111193595t^{13} - 100853469064718537930t^{12} - 37307260370135518113t^{11} - 7195002022227921963t^{10} - 1033575393870227586t^{9} - 104618525734483938t^{8} - 5239182752025961t^{7} + 152945038508838t^{6} + 42759361081413t^{5} + 4032890764132t^{4} + 255479378482t^{3} + 6508477682t^{2} + 23227134t + 448400 \right)D^{4} + t \left( 281126462874102551481720t^{21} + 1309401420647103240983280t^{20} + 2655174968707606768651935t^{19} + 3148757551655956948695376t^{18} + 2466825336079265820509987t^{17} + 1355759590453103427766776t^{16} + 535981337879817252914521t^{15} + 152281317215725752153756t^{14} + 30328202871758399680020t^{13} + 3985309475862428974975t^{12} + 276459648177650669704t^{11} - 15120967947908397004t^{10} - 9725626937669274118t^{9} - 2156270260223853332t^{8} - 294102674736433142t^{7} - 24342411886858488t^{6} - 1170707779196327t^{5} - 32737694919759t^{4} - 711884593294t^{3} - 19926394302t^{2} - 251492090t - 185850 \right)D^{3} + 2 t ^{2} \left( 310577806603770437827424t^{20} + 1429947978460577661807718t^{19} + 2858663517325070445001161t^{18} + 3341420205473379092152946t^{17} + 2587110894201227265175758t^{16} + 1412040640953097417034320t^{15} + 558728518548914567194316t^{14} + 161330746977253052822546t^{13} + 33871931562846492668540t^{12} + 5183414890526111972058t^{11} + 587114181524190061613t^{10} + 44374957179667633993t^{9} - 846450385466966780t^{8} - 1001927867086362043t^{7} - 162389283340672101t^{6} - 13162534600234485t^{5} - 588523135427773t^{4} - 16746807783489t^{3} - 455656936116t^{2} - 11992377130t - 97819296 \right)D^{2} + 4 t ^{2} \left( 168675877724461530889032t^{20} + 769991442148923595998738t^{19} + 1522563783947990138567584t^{18} + 1758967520398290092830750t^{17} + 1347876048352618376211361t^{16} + 730003374523930892375528t^{15} + 287763048459982003978380t^{14} + 83386066983125866235912t^{13} + 17864491355523240966188t^{12} + 2894592300812402195061t^{11} + 371585680177612516495t^{10} + 36926194257654990753t^{9} + 1757396306698388949t^{8} - 212887478931826599t^{7} - 48122524282161482t^{6} - 3890211849549702t^{5} - 164902751098618t^{4} - 4917421213534t^{3} - 151532318696t^{2} - 3540595720t - 20447040 \right)D + 16 t ^{2} \left( 17211824257598115396840t^{20} + 78114304293932776600710t^{19} + 153285239493479478747738t^{18} + 175593495559242593535672t^{17} + 133507046459175552364418t^{16} + 71842033774135396515380t^{15} + 28193975264187481715398t^{14} + 8163866341445291346532t^{13} + 1762821034657993364923t^{12} + 293062162240803644292t^{11} + 39600740916743341464t^{10} + 4282276096612722788t^{9} + 276962303972613250t^{8} - 6014951314206663t^{7} - 2969074736894699t^{6} - 244902785966987t^{5} - 10116671661131t^{4} - 326145480423t^{3} - 10781474584t^{2} - 216180130t - 896800 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 8.561717 λ ≈ 8.561717; multiplicity 1; Jordan blocks 1 λ ≈ -1.37806 - 6.205561i; multiplicity 1; Jordan blocks 1 λ ≈ -1.37806 + 6.205561i; multiplicity 1; Jordan blocks 1 λ ≈ -3.234552 - 1.940705i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -3.234552 + 1.940705i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -2.902798 - 0.6839577i; multiplicity 1; Jordan blocks 1 λ ≈ -2.902798 + 0.6839577i; multiplicity 1; Jordan blocks 1 λ ≈ 2.799605; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -1.3305; multiplicity 3; Jordan blocks 1, 1, 1 3 3 3 3
Sector dimension
17
Spectral radius
$8.56171705$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.561717$11
$-1.37806 - 6.205561\mathrm{i}$11
$-1.37806 + 6.205561\mathrm{i}$11
$-3.234552 - 1.940705\mathrm{i}$31, 1, 1
$-3.234552 + 1.940705\mathrm{i}$31, 1, 1
$-2.902798 - 0.6839577\mathrm{i}$11
$-2.902798 + 0.6839577\mathrm{i}$11
$2.799605$31, 1, 1
$-1.3305$31, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{4} + 5 \lambda^{3} + \lambda^{2} - 45 \lambda - 53\right)^{3} \left(\lambda^{5} - 8 \lambda^{3} - 300 \lambda^{2} - 1859 \lambda - 3077\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{4} + 5 \lambda^{3} + \lambda^{2} - 45 \lambda - 53\right) \left(\lambda^{5} - 8 \lambda^{3} - 300 \lambda^{2} - 1859 \lambda - 3077\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 2 & 0 & 0 & 3 & 0 & 3 & 3 & 9 & 0 & 4 & 4 & 4 & 5 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 3 & 3 & 0 & 4 \\ 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 3 & 0 & 3 & 4 \\ 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 3 & 3 & 4 \\ 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 \\ 0 & 2 & 0 & 0 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 \\ 0 & 0 & 2 & 0 & 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 1 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & -2 & -2 & -7 & -3 & -2 & -2 & -2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 & 0 & 3 & 0 & 4 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0 & 0 & 3 & 4 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 4 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$