Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-183-43-c

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × Gr(2, 5)
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_5node 6: fundamental weight ω_6 (marked)node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
2 copies of E^(w6) + E^(w3 + w5) + E^(w1 + w5)
Bundle
2 copies of O_3(1) + O_2(1) box U_3^v + O_1(1) box U_3^v
ZL62 label
202041.55wEw3
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 43
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 183
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 183
$\mathrm{c}_1^2\mathrm{c}_2$ = 138
$\mathrm{c}_2^2$ = 105
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 8 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 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 32 0 0 x1 + 4 28
$\mathrm{HH}^{ 4 }$ = 43 0 0 0 0 43
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 38t + 105t^{2} + 38t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(406)
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( 3t + 1 \right) \left( 89t^{4} + 59t^{3} + 17t^{2} + 5t + 1 \right) \left( 877t^{6} + 1233t^{5} + 2300t^{4} + 1011t^{3} + 121t^{2} - 4t - 1 \right) \left( 2556975395781128796t^{20} + 12951201943384019168t^{19} + 29241251272929154260t^{18} + 40996195725162660788t^{17} + 41126126843200719263t^{16} + 31749086376093585392t^{15} + 19536038700312931615t^{14} + 9712631066100017501t^{13} + 3914975807301487961t^{12} + 1279285127427958552t^{11} + 338319893825827811t^{10} + 72082785309194635t^{9} + 12240674107552222t^{8} + 1622032479707082t^{7} + 161869715787404t^{6} + 11554733966168t^{5} + 559280941884t^{4} + 19202059760t^{3} + 633570644t^{2} + 13672264t - 295728 \right)D^{6} + \left( 12573514835714980092593844t^{31} + 91168863851023834133336940t^{30} + 323134679315518234568800356t^{29} + 755267560935859720344621008t^{28} + 1300555245746927251411547285t^{27} + 1743838807068509268200275375t^{26} + 1885139877043492149134444323t^{25} + 1683864511959399227587952562t^{24} + 1264417535008429860880181866t^{23} + 807253817744392716648191975t^{22} + 441219586121372082424173660t^{21} + 207295833643410089844157107t^{20} + 83934738682733296742353358t^{19} + 29345919784110332256609615t^{18} + 8874014404259471868843450t^{17} + 2324298900871057224883127t^{16} + 528029609641781016864979t^{15} + 104203366874494065354772t^{14} + 17902266411332568339841t^{13} + 2687295560717166298640t^{12} + 354630454652632457986t^{11} + 41527001651934767817t^{10} + 4359195623971108056t^{9} + 411010911810753346t^{8} + 34069121649344240t^{7} + 2350964827340848t^{6} + 124384541204844t^{5} + 4622476035488t^{4} + 121298720448t^{3} + 3083570740t^{2} + 58237792t - 887184 \right)D^{5} + \left( 104779290297624834104948700t^{31} + 741603981762961288610914500t^{30} + 2513987436975540145822971772t^{29} + 5552482459736561028054407348t^{28} + 9011409354863502157169235991t^{27} + 11431071952511306004920484493t^{26} + 11765735253551389371978422881t^{25} + 10070300385862234672740104063t^{24} + 7280746266424114538379017349t^{23} + 4486950784935064188412682331t^{22} + 2367580098950638274178097397t^{21} + 1071554730083111164560394941t^{20} + 416193589918989067744567200t^{19} + 138693966285271289810663185t^{18} + 39615427101794888891626339t^{17} + 9675666869733544317818695t^{16} + 2011295758856491614923491t^{15} + 352901441967459757259076t^{14} + 51532911602095474860190t^{13} + 6108317977084843498649t^{12} + 558540930190409957393t^{11} + 34152890055489324383t^{10} + 429524224317342405t^{9} - 200395487918578642t^{8} - 29514612910019610t^{7} - 2483825505676084t^{6} - 140389596121388t^{5} - 5214144020348t^{4} - 123848835212t^{3} - 2694985088t^{2} - 58587656t + 591456 \right)D^{4} + t \left( 440073019250024303240784540t^{30} + 3054886107319054225915092900t^{29} + 10011781710460114847529174780t^{28} + 21183713200779188946987224080t^{27} + 32846683529201612967133221655t^{26} + 39888394794982223441803040205t^{25} + 39472006532482024229650326397t^{24} + 32626582561106813464285198998t^{23} + 22861569437568447496441908634t^{22} + 13684518650166421244329446905t^{21} + 7019427109933530788618738076t^{20} + 3087706368504383525204265405t^{19} + 1164391793971252357622901392t^{18} + 376118216728706905070265393t^{17} + 103900172244131471562143522t^{16} + 24472537285553282253355085t^{15} + 4889914602915551418636355t^{14} + 822450613315521273055114t^{13} + 115209426326886680561003t^{12} + 13268239551832598271560t^{11} + 1240155592285795379522t^{10} + 93558317313343853309t^{9} + 5817463627693545148t^{8} + 320731970422151594t^{7} + 16921962044255456t^{6} + 770029611125540t^{5} + 19591739063436t^{4} - 279956680732t^{3} - 31556197864t^{2} - 413087672t + 995456 \right)D^{3} + 2 t \left( 486175906980979230246961968t^{30} + 3324333774013643165965684680t^{29} + 10625028295012651861799656724t^{28} + 21785381455766780653086610644t^{27} + 32663238759083638141792636580t^{26} + 38390013886383568220825124368t^{25} + 36851314615472978388247077645t^{24} + 29615625933547686404202495663t^{23} + 20210335221852829018762158790t^{22} + 11794051404792520065565728096t^{21} + 5901169820118659838834962572t^{20} + 2532613096686630429732642427t^{19} + 931747880999216248318210706t^{18} + 293495685075263789342017002t^{17} + 78983445966423221635360395t^{16} + 18089410592020789955346249t^{15} + 3503686927452570670604737t^{14} + 568640314106931116469511t^{13} + 76418946424962540859598t^{12} + 8394578831788786188049t^{11} + 746956324366335393929t^{10} + 54228606628402917699t^{9} + 3357616140369770168t^{8} + 190594429842922058t^{7} + 9766342972441048t^{6} + 356814060745208t^{5} + 3338959892056t^{4} - 381614675300t^{3} - 15569175342t^{2} - 84573816t + 147864 \right)D^{2} + 4 t ^{2} \left( 264043811550014581944470724t^{29} + 1785323219778059546485960740t^{28} + 5604488340506388977083587216t^{27} + 11239256322741097005102072708t^{26} + 16457256669141044399961830793t^{25} + 18897285645521034881667537183t^{24} + 17738857409340817680073071582t^{23} + 13950005420376588682245974053t^{22} + 9317494352726561786726557394t^{21} + 5321821772166579668516086459t^{20} + 2606383384342370230189328438t^{19} + 1095175988548364321582784166t^{18} + 394632252347233067275641030t^{17} + 121790792931004825578523199t^{16} + 32112983488032552587602257t^{15} + 7202481617656085994150394t^{14} + 1364501470169650546824798t^{13} + 216219508141524655397343t^{12} + 28323229409069739368263t^{11} + 3034441790557569972555t^{10} + 265339074214902902311t^{9} + 19339746781377004154t^{8} + 1238761340165965278t^{7} + 72064191881525672t^{6} + 3423893056989094t^{5} + 91890582668740t^{4} - 1080603844638t^{3} - 168507981912t^{2} - 4170822714t - 7167744 \right)D + 48 t ^{3} \left( 8981082025510700066138460t^{28} + 60262612484060596925537100t^{27} + 186917460844280202006978696t^{26} + 369387280580028287576263764t^{25} + 532505092368613203980133679t^{24} + 602045902075341769800831401t^{23} + 556573432712911419390517192t^{22} + 430989321333362531676824805t^{21} + 283301200985503140185999706t^{20} + 159151258510211547481591459t^{19} + 76639156404004062483856718t^{18} + 31665619097954196107234055t^{17} + 11224029642872851676183264t^{16} + 3409060760054026677661943t^{15} + 884983792181929893305355t^{14} + 195442909957772807153430t^{13} + 36449894243491277784167t^{12} + 5684188524855394101403t^{11} + 733191535496406144479t^{10} + 77644461698907455510t^{9} + 6788091610376467851t^{8} + 505736506858468032t^{7} + 33740072097265224t^{6} + 1998738394180812t^{5} + 89540373474183t^{4} + 1857147101880t^{3} - 56470328136t^{2} - 4362500160t - 77850396 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × Gr(2, 5)

|λ| = 12.78958 λ ≈ 12.78958; multiplicity 1; Jordan blocks 1 λ ≈ -6.490324 - 0.509338i; multiplicity 1; Jordan blocks 1 λ ≈ -6.490324 + 0.509338i; multiplicity 1; Jordan blocks 1 λ ≈ 0.206542 - 3.383479i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.206542 + 3.383479i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3.387373; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.706542 - 0.6481502i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.706542 + 0.6481502i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.2107823 - 0.6581711i; multiplicity 1; Jordan blocks 1 λ ≈ -0.2107823 + 0.6581711i; multiplicity 1; Jordan blocks 1 2 2 2 2
Sector dimension
15
Spectral radius
$12.7895846$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.78958$11
$-6.490324 - 0.509338\mathrm{i}$11
$-6.490324 + 0.509338\mathrm{i}$11
$0.206542 - 3.383479\mathrm{i}$21, 1
$0.206542 + 3.383479\mathrm{i}$21, 1
$-3.387373$11
$-3$11
$-2.706542 - 0.6481502\mathrm{i}$21, 1
$-2.706542 + 0.6481502\mathrm{i}$21, 1
$-0.2107823 - 0.6581711\mathrm{i}$11
$-0.2107823 + 0.6581711\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 5 \lambda^{3} + 17 \lambda^{2} + 59 \lambda + 89\right)^{2} \left(\lambda^{6} + 4 \lambda^{5} - 121 \lambda^{4} - 1011 \lambda^{3} - 2300 \lambda^{2} - 1233 \lambda - 877\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 5 \lambda^{3} + 17 \lambda^{2} + 59 \lambda + 89\right) \left(\lambda^{6} + 4 \lambda^{5} - 121 \lambda^{4} - 1011 \lambda^{3} - 2300 \lambda^{2} - 1233 \lambda - 877\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 8 & 9 & 15 & 6 & 15 & 6 & 9 & 12 & 56 & 20 & 28 & 50 \\ 1 & -1 & 0 & 2 & 4 & 4 & 0 & 6 & 2 & 2 & 4 & 6 & 3 & 3 & 8 \\ 1 & 0 & -1 & 2 & 4 & 6 & 2 & 4 & 0 & 2 & 4 & 6 & 0 & 6 & 8 \\ 1 & 1 & 1 & -1 & 0 & 4 & 2 & 4 & 2 & 4 & 4 & 15 & 6 & 6 & 16 \\ 0 & 1 & 1 & 0 & -2 & 2 & 0 & 2 & 0 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & -2 & 0 & 0 & 0 & 1 & 1 & 4 & 2 & 2 & 0 \\ 0 & 1 & 0 & 0 & 0 & 2 & -1 & 0 & 0 & 0 & 0 & 6 & 2 & 2 & 6 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & -2 & 0 & 1 & 1 & 4 & 0 & 4 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & -1 & 0 & 0 & 6 & 4 & 0 & 6 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & -1 & -2 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 3 & 4 & 4 & -6 & -2 & -4 & -5 & -3 & -2 & 2 & 0 \\ 0 & 0 & 0 & 0 & -\frac{3}{2} & -\frac{9}{2} & -\frac{11}{2} & 13 & 5 & \frac{13}{2} & \frac{17}{2} & \frac{9}{2} & 2 & -1 & 4 \\ 0 & 0 & 0 & 0 & -\frac{3}{2} & \frac{1}{2} & -\frac{5}{2} & 8 & 2 & \frac{13}{2} & \frac{17}{2} & \frac{9}{2} & 3 & -2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{9}{2} & \frac{3}{2} & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$