Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-162-39-d

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ6P6
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_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(w3 + w5) + E^(w4 + w5) + E^(w1 + w5) + E^(w2 + w5)
Bundle
O_2(1) box O_3(1) + Q_2 box O_3(1) + O_1(1) box O_3(1) + Q_1 box O_3(1)
ZL62 label
2020600.wOwIwBw3
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 162
$\chi(X, \mathrm{T}_X)$ = -10

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 162
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 111
$\mathrm{c}_1\mathrm{c}_3$ = 42
$\mathrm{c}_4$ = 21
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 13 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 + 10 0 x0 + 10 x1
$\mathrm{HH}^{ 3 }$ = x1 + 29 0 0 x1 + 5 24
$\mathrm{HH}^{ 4 }$ = 39 0 0 0 0 39
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 92t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(419)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(486)

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) \left( t + 1 \right) ^{2} \left( 29t^{3} + 19t^{2} + 2t + 1 \right) \left( 1816t^{4} + 1080t^{3} + 142t^{2} - 3t - 1 \right) \left( 1102336952962688t^{16} + 5679893856528896t^{15} + 13597553797891200t^{14} + 19969020035844416t^{13} + 19886187245902304t^{12} + 14111425153949568t^{11} + 7361272353075252t^{10} + 2889351616817592t^{9} + 867615889441629t^{8} + 199926399522199t^{7} + 34512892477987t^{6} + 4183743185905t^{5} + 316315249170t^{4} + 12431800546t^{3} + 306936908t^{2} + 13827076t - 181440 \right)D^{6} + \left( t + 1 \right) \left( 2438245878214734034944t^{25} + 18124888012277766456320t^{24} + 63741993168297540018688t^{23} + 140864356668314561463296t^{22} + 218773682523196252216704t^{21} + 253084649386256789649024t^{20} + 225864868844414851508160t^{19} + 159221059064780643948896t^{18} + 90206628687421679785280t^{17} + 41623456157142604661640t^{16} + 15799394448130676474024t^{15} + 4965105478692525844804t^{14} + 1294617881639307668539t^{13} + 279439425239732831957t^{12} + 49649814126446957168t^{11} + 7218943495669866229t^{10} + 859271660951508165t^{9} + 85425714067773493t^{8} + 7510576640279412t^{7} + 625492737636623t^{6} + 47233291019108t^{5} + 2605651955878t^{4} + 78240804620t^{3} + 1393736556t^{2} + 57485584t - 544320 \right)D^{5} + \left( 20318715651789450291200t^{26} + 163737452911233555706880t^{25} + 624313552195976650680832t^{24} + 1496947446725993927144960t^{23} + 2526310039027046402261120t^{22} + 3182050716022733068092160t^{21} + 3098318072820600063949120t^{20} + 2387081799860855092159264t^{19} + 1479857734911137231048224t^{18} + 747663959983475892511816t^{17} + 310777196476348795256488t^{16} + 106914222326156942631480t^{15} + 30481112663554808338057t^{14} + 7167478030328642933683t^{13} + 1373559900825114052093t^{12} + 210278102308426500318t^{11} + 24959182244187126769t^{10} + 2189956430295712166t^{9} + 127878938190995779t^{8} + 2912763394857840t^{7} - 313627988937514t^{6} - 45471865850177t^{5} - 2999907001452t^{4} - 95897103518t^{3} - 1271696928t^{2} - 51680204t + 362880 \right)D^{4} + t \left( 85338605737515691223040t^{25} + 662544714845707904291840t^{24} + 2438310464179812234150400t^{23} + 5650802412902202362765824t^{22} + 9221784969150224438508672t^{21} + 11228844765835320015554048t^{20} + 10562306134554018667510720t^{19} + 7855994246536795711690080t^{18} + 4699430810004696654239648t^{17} + 2290409835491751902988792t^{16} + 918101634800552002538128t^{15} + 304239239403993642472460t^{14} + 83298557395748375314757t^{13} + 18697555990966103179346t^{12} + 3387210643780279428751t^{11} + 483789478178312872879t^{10} + 52882002417296429200t^{9} + 4286876721463763100t^{8} + 254245275350625379t^{7} + 11814653387187645t^{6} + 521878961540463t^{5} + 20829186335382t^{4} + 332148962410t^{3} - 16998181124t^{2} - 295950036t + 613760 \right)D^{3} + 2 t \left( 94278840624303049351168t^{25} + 710694236114251290609664t^{24} + 2542917089063917881582592t^{23} + 5735229269545599131659264t^{22} + 9110067176643184703975680t^{21} + 10792006403667927683252992t^{20} + 9869094709836525742903680t^{19} + 7131975284748679440137376t^{18} + 4143943934230790876032400t^{17} + 1961658726587696204241680t^{16} + 763616808333219713440724t^{15} + 245493575173738482416176t^{14} + 65025371893464097567584t^{13} + 14042984723104562949730t^{12} + 2426347340204862612805t^{11} + 326599736469059308454t^{10} + 33183833256952302439t^{9} + 2478688326878639564t^{8} + 138517612788748927t^{7} + 6651823700545582t^{6} + 308647043577301t^{5} + 9450655563296t^{4} - 15074286676t^{3} - 10449734880t^{2} - 56949156t + 90720 \right)D^{2} + 4 t ^{2} \left( 51203163442509414733824t^{24} + 377519088944889378059264t^{23} + 1322283312719744056652288t^{22} + 2921021135839936792563200t^{21} + 4544175536525831395372672t^{20} + 5268980954758486578575872t^{19} + 4712714233749704900655680t^{18} + 3329025001351290739573120t^{17} + 1890194663154583031862192t^{16} + 874313304217186005190216t^{15} + 332481006320480940004836t^{14} + 104308101835059341700060t^{13} + 26889997957293256610281t^{12} + 5624681879224904675406t^{11} + 934630715243696386030t^{10} + 119947759683972091331t^{9} + 11535228172987697429t^{8} + 819500723954351282t^{7} + 45549716255185968t^{6} + 2336328305929001t^{5} + 106146575767594t^{4} + 2167160336458t^{3} - 57289808450t^{2} - 3054049132t - 4271616 \right)D + 192 t ^{3} \left( 435401049681202506240t^{23} + 3161586665074289258240t^{22} + 10910572631718534717824t^{21} + 23753686360555525778432t^{20} + 36409802043193345639968t^{19} + 41571000676102034896416t^{18} + 36587111730427222407712t^{17} + 25416548237295106043888t^{16} + 14187295036189277837872t^{15} + 6450181555999926568224t^{14} + 2410063132703396402522t^{13} + 742109467187494922719t^{12} + 187319537423318308389t^{11} + 38206810557537880994t^{10} + 6155008576350216987t^{9} + 760901430435136831t^{8} + 70248285580743035t^{7} + 4850073037290244t^{6} + 275429947795055t^{5} + 14984759857424t^{4} + 651170599012t^{3} + 8576764992t^{2} - 511589952t - 14696640 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ6P6

|λ| = 13.7457 λ ≈ 13.7457; multiplicity 1; Jordan blocks 1 λ ≈ -7.107661 - 1.300909i; multiplicity 1; Jordan blocks 1 λ ≈ -7.107661 + 1.300909i; multiplicity 1; Jordan blocks 1 λ ≈ -0.2092939 - 4.277176i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.2092939 + 4.277176i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.530377; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.581412; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 2 2 2 2 2
Sector dimension
14
Spectral radius
$13.7456989$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.7457$11
$-7.107661 - 1.300909\mathrm{i}$11
$-7.107661 + 1.300909\mathrm{i}$11
$-0.2092939 - 4.277176\mathrm{i}$21, 1
$-0.2092939 + 4.277176\mathrm{i}$21, 1
$-2.530377$11
$-2$21, 1
$-1.581412$21, 1
$-1$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right)^{2} \left(\lambda^{3} + 2 \lambda^{2} + 19 \lambda + 29\right)^{2} \left(\lambda^{4} + 3 \lambda^{3} - 142 \lambda^{2} - 1080 \lambda - 1816\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{3} + 2 \lambda^{2} + 19 \lambda + 29\right) \left(\lambda^{4} + 3 \lambda^{3} - 142 \lambda^{2} - 1080 \lambda - 1816\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 8 & 12 & 18 & 9 & 18 & 9 & 24 & 80 & 32 & 32 & 160 \\ 1 & -1 & 0 & 2 & 4 & 4 & 0 & 8 & 4 & 8 & 6 & 3 & 0 & 32 \\ 1 & 0 & -1 & 2 & 4 & 8 & 4 & 4 & 0 & 8 & 6 & 0 & 3 & 32 \\ 1 & 1 & 1 & -1 & 0 & 4 & 2 & 4 & 2 & 8 & 24 & 12 & 12 & 48 \\ 0 & 1 & 1 & 0 & -2 & 2 & 0 & 2 & 0 & 4 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & -2 & -1 & 0 & 0 & 0 & 4 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 8 & 4 & 4 & 12 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & -2 & -1 & 0 & 4 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 8 & 4 & 4 & 12 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 12 \\ 0 & 0 & 0 & 0 & 1 & \frac{7}{2} & \frac{1}{2} & \frac{7}{2} & \frac{1}{2} & 7 & 1 & \frac{3}{2} & \frac{3}{2} & 8 \\ 0 & 0 & 0 & 0 & 1 & -3 & -1 & 1 & 2 & -4 & 0 & -2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 2 & -3 & -1 & -4 & 0 & -1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{4} & \frac{3}{4} & 0\end{pmatrix}\notag\end{equation}$$