Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-241-54-d

·

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

Defining data
Ambient
Fl(1, 2, 5) × ℙ3P3
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w1) + 2 copies of E^(w2) + E^(w4 + w5)
Bundle
O_1(1,0) + 2 copies of O_1(0,1) + wedge^2 Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1)
ZL62 label
3042.443w5
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 241
$\chi(X, \mathrm{T}_X)$ = 0

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 241
$\mathrm{c}_1^2\mathrm{c}_2$ = 154
$\mathrm{c}_2^2$ = 102
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 15
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 7 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 0 x0 x1
$\mathrm{HH}^{ 3 }$ = x1 + 40 0 0 x1 - 12 52
$\mathrm{HH}^{ 4 }$ = 54 0 0 0 0 54
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 49t + 141t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(211)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( 17t^{2} + 2t + 1 \right) \left( 1873t^{7} + 4673t^{6} + 3818t^{5} + 1860t^{4} + 638t^{3} + 61t^{2} - 5t - 1 \right) \left( 1455751602691322t^{17} + 3924817338293200t^{16} + 2814417447699096t^{15} - 2617926209216022t^{14} - 7172035762745982t^{13} - 7038962123848718t^{12} - 4266258434962821t^{11} - 1724696986019184t^{10} - 386664606367189t^{9} - 10179933306813t^{8} + 10845290401482t^{7} + 36430178980t^{6} - 366875904292t^{5} + 5280300310t^{4} + 3138354587t^{3} - 270022222t^{2} - 8891t - 21931 \right)D^{6} + \left( 973404322407182059842t^{27} + 5596029716133478765672t^{26} + 13202412006369201410760t^{25} + 14477128294401027763116t^{24} + 159202453910733902340t^{23} - 24608130864929117130050t^{22} - 43292690397175233779551t^{21} - 45362974316143980178166t^{20} - 34151982110783563163017t^{19} - 19505623605298221630350t^{18} - 8530761685918713634359t^{17} - 2806466590582081723643t^{16} - 657899766026373148287t^{15} - 93672426022502803706t^{14} - 2978754133927313570t^{13} + 1045432360217208053t^{12} - 133238061887162225t^{11} - 111518318972403708t^{10} - 16333970798669904t^{9} + 12654732052465t^{8} + 181157950638959t^{7} - 8395077349964t^{6} - 3770470749311t^{5} + 26081004277t^{4} + 15535081860t^{3} - 1083570582t^{2} + 324223t - 43862 \right)D^{5} + \left( 8111702686726517165350t^{27} + 43001323150786217934840t^{26} + 91380034630498054095192t^{25} + 79861036340989135033754t^{24} - 44641402887350690021940t^{23} - 221744182933720914113476t^{22} - 329242421270382084957955t^{21} - 312646192338614672449494t^{20} - 215868026041343147431207t^{19} - 112090044703025635085819t^{18} - 43339428296652499539743t^{17} - 11908640305568643723559t^{16} - 2051359093937723199641t^{15} - 116003084636030291431t^{14} + 32441890116718075391t^{13} + 4709677860570639231t^{12} - 1294098925665179006t^{11} - 384902844085353712t^{10} - 16669865833761390t^{9} + 2647192651407644t^{8} - 43319627032377t^{7} - 36002077808351t^{6} + 576468723606t^{5} + 27901479901t^{4} - 21405420987t^{3} + 1086080189t^{2} + 2483427t + 21931 \right)D^{4} + t \left( 34069151284251372094470t^{26} + 168619106437225968455320t^{25} + 326192324815250404782520t^{24} + 220409925371921681376076t^{23} - 296791230950652909193244t^{22} - 936498603735582177799570t^{21} - 1252769170913438061831497t^{20} - 1111903895738310075769882t^{19} - 721746450556869563148843t^{18} - 348238543404971133205298t^{17} - 121059902057963559607667t^{16} - 27861378595306822538389t^{15} - 3301548934273206109009t^{14} + 80149616891261308468t^{13} + 44875326376125015844t^{12} - 19992541276204904865t^{11} - 9248291824077114665t^{10} - 1351774109072615076t^{9} - 30140204722578560t^{8} + 7110523033524429t^{7} - 485657702612855t^{6} - 147046531629330t^{5} - 9790737517187t^{4} - 100295668489t^{3} - 1783803214t^{2} - 33636114t - 9373 \right)D^{3} + 2 t ^{2} \left( 37638300466411039647224t^{25} + 176152560367299564819192t^{24} + 314099115513704380763960t^{23} + 157854577071126952096436t^{22} - 397428423909257347821808t^{21} - 1015931857475246187283202t^{20} - 1267555753249154649246432t^{19} - 1072997307779553433698301t^{18} - 666123258301666174666791t^{17} - 303529889704230077023077t^{16} - 96343275691857973373415t^{15} - 18716540569504718726059t^{14} - 1392304888393970827166t^{13} + 117501392410695038257t^{12} - 45047360585455289770t^{11} - 39296169515728163584t^{10} - 9172414625637006825t^{9} - 848367059042371286t^{8} + 11642055202389157t^{7} + 3613352331987589t^{6} - 927253988505164t^{5} - 115791997217005t^{4} - 4722984291116t^{3} - 29003880913t^{2} - 1286256206t - 12196287 \right)D^{2} + 4 t ^{2} \left( 20441490770550823256682t^{25} + 91636787092729555153072t^{24} + 152843235810466812698960t^{23} + 54732233113078560449540t^{22} - 237824884652632344025750t^{21} - 539218055826413887938478t^{20} - 641610741951026751913185t^{19} - 525482406082039200180959t^{18} - 315899296107042827132508t^{17} - 137699391799427591657022t^{16} - 40524979828611467424900t^{15} - 6753019690301925420866t^{14} - 286943859402424269106t^{13} + 7845513591367093632t^{12} - 57887115488938840514t^{11} - 24401612766815738244t^{10} - 4122999245329321446t^{9} - 250537753830039968t^{8} + 11938833826989498t^{7} + 262086544160564t^{6} - 517671682594062t^{5} - 43157958098996t^{4} - 1182323930059t^{3} - 8311914379t^{2} - 391196010t - 2280824 \right)D + 8 t ^{2} \left( 4171732810316494542180t^{25} + 18145427291833668636480t^{24} + 28802052243938838043776t^{23} + 7150672113867494650344t^{22} - 51114261192183583112426t^{21} - 107940020880372170356574t^{20} - 124502857025697367560870t^{19} - 99744759773191087117562t^{18} - 58646851965629105102890t^{17} - 24746432993204145678341t^{16} - 6868687416317523150984t^{15} - 1006650003761739794827t^{14} - 22574791825022377827t^{13} - 9656883530278611738t^{12} - 15741766778921662625t^{11} - 5100853011693559441t^{10} - 700970634122873957t^{9} - 27367163428786245t^{8} + 2424023907683802t^{7} - 149514772803111t^{6} - 98398186029917t^{5} - 6322228787034t^{4} - 122699016719t^{3} - 1396204279t^{2} - 46904039t - 175448 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5) × ℙ3P3

|λ| = 9.937049 λ ≈ 9.937049; multiplicity 1; Jordan blocks 1 λ ≈ -5.888357 - 2.889717i; multiplicity 1; Jordan blocks 1 λ ≈ -5.888357 + 2.889717i; multiplicity 1; Jordan blocks 1 λ ≈ -1 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.451335 - 1.967762i; multiplicity 1; Jordan blocks 1 λ ≈ -0.451335 + 1.967762i; multiplicity 1; Jordan blocks 1 λ ≈ -1.575341; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.6823248; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
13
Spectral radius
$9.93704895$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.937049$11
$-5.888357 - 2.889717\mathrm{i}$11
$-5.888357 + 2.889717\mathrm{i}$11
$-1 - 4\mathrm{i}$21, 1
$-1 + 4\mathrm{i}$21, 1
$-0.451335 - 1.967762\mathrm{i}$11
$-0.451335 + 1.967762\mathrm{i}$11
$-1.575341$11
$-1$21, 1
$-0.6823248$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{2} + 2 \lambda + 17\right)^{2} \left(\lambda^{7} + 5 \lambda^{6} - 61 \lambda^{5} - 638 \lambda^{4} - 1860 \lambda^{3} - 3818 \lambda^{2} - 4673 \lambda - 1873\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} + 2 \lambda + 17\right) \left(\lambda^{7} + 5 \lambda^{6} - 61 \lambda^{5} - 638 \lambda^{4} - 1860 \lambda^{3} - 3818 \lambda^{2} - 4673 \lambda - 1873\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 4 & 2 & 3 & 9 & 6 & 9 & 6 & 12 & 12 & 16 & 20 \\ 1 & -1 & 1 & 0 & 2 & 2 & 0 & 6 & 4 & 0 & 0 & 0 & 8 \\ 1 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 6 & 6 & 6 & 0 \\ 1 & 0 & 1 & -1 & 2 & 2 & 4 & 2 & 4 & 0 & 0 & 0 & 8 \\ 0 & 1 & 2 & -1 & -1 & 1 & 0 & 0 & -2 & -4 & -2 & -4 & 0 \\ 0 & 1 & 0 & 1 & 1 & -2 & 0 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 2 & 4 & 4 & 0 \\ 0 & 0 & 1 & 2 & 0 & 1 & 0 & -1 & 0 & 4 & 2 & 4 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 1 & 5 & 5 & 2 & 1 & 0 & 1 & 2 & 4 \\ 0 & 0 & 0 & 0 & 1 & 5 & 1 & 6 & 1 & 1 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 1 & -\frac{9}{2} & -\frac{5}{2} & -\frac{3}{2} & \frac{5}{2} & -\frac{1}{2} & -\frac{1}{2} & -3 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{5}{2} & 3 & 0\end{pmatrix}\notag\end{equation}$$