Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-112-30-e

·

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

Defining data
Ambient
ℙ2P2 × ℙ3P3
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_5ω1ω2ω3ω4ω5
Bundle
E^(2w1 + 2w3)
Bundle
O_1(2) box O_2(2)
ZL62 label
2030.x318
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 112
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 112
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 4
$\mathrm{c}_4$ = 52
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 40 3 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 36 0 36 0
$\mathrm{HH}^{ 3 }$ = 21 0 0 3 18
$\mathrm{HH}^{ 4 }$ = 36 0 0 6 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 60t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(362)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(439)

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) ^{3} \left( 152t^{3} + 84t^{2} + 14t - 1 \right) \left( 392t^{3} + 172t^{2} + 10t + 1 \right) \left( 6483656711680t^{13} + 21238264866560t^{12} + 30531820232192t^{11} + 28775310032128t^{10} + 20764091467648t^{9} + 11191576599264t^{8} + 4154141896144t^{7} + 1000420046288t^{6} + 150609298914t^{5} + 14087527143t^{4} + 887901031t^{3} + 39039187t^{2} + 206625t + 1386 \right)D^{6} + \left( 2t + 1 \right) ^{2} \left( 16225532463367127040t^{20} + 72179261216439828480t^{19} + 147035258470998130688t^{18} + 191622426314342178816t^{17} + 184839248004582920192t^{16} + 139499870826009884672t^{15} + 82448122884734045184t^{14} + 37462210038374697472t^{13} + 12856640833893242624t^{12} + 3290345608448015488t^{11} + 622394083052769344t^{10} + 86639997940213728t^{9} + 8924418564155760t^{8} + 695941415202024t^{7} + 42547417387556t^{6} + 2080320383326t^{5} + 86969694244t^{4} + 3934375135t^{3} + 153851662t^{2} + 603243t + 2772 \right)D^{5} + \left( 2t + 1 \right) \left( 270425541056118784000t^{21} + 1269280918473182412800t^{20} + 2737527157803315789824t^{19} + 3780518358759942832128t^{18} + 3866084246051450642432t^{17} + 3103132088425968635904t^{16} + 1964383911176398837760t^{15} + 965747974965559884800t^{14} + 363121875498436831744t^{13} + 103347542256539490560t^{12} + 22124201386321794176t^{11} + 3553206585423140672t^{10} + 429282546751376544t^{9} + 39267806167342128t^{8} + 2682840591307528t^{7} + 122605747176308t^{6} + 2326601113688t^{5} - 58396588707t^{4} - 3558778465t^{3} - 137396065t^{2} - 489843t - 1386 \right)D^{4} + t \left( 2271574544871397785600t^{21} + 11342878030239861964800t^{20} + 26123882948546812313600t^{19} + 38488571472681689645056t^{18} + 41887146168002024194048t^{17} + 35821858608906062831616t^{16} + 24332510415641671856128t^{15} + 12981378065242771509248t^{14} + 5367028314303633510400t^{13} + 1703613896817696202752t^{12} + 412882330195182988800t^{11} + 76219290724113553536t^{10} + 10737464728715146816t^{9} + 1163035368182037536t^{8} + 97518812018008240t^{7} + 6190551541084640t^{6} + 268634769093340t^{5} + 6285503260742t^{4} + 54815124202t^{3} + 361616457t^{2} + 2613390t + 2673 \right)D^{3} + 2 t ^{2} \left( 2509549021000782315520t^{20} + 12146706967058914017280t^{19} + 27061905829494511239168t^{18} + 38782949049796344774656t^{17} + 41371657826877123170304t^{16} + 34752114292502106613760t^{15} + 23053835274781901408256t^{14} + 11895587054485089771520t^{13} + 4710656008132398084608t^{12} + 1420811707334910033664t^{11} + 325390342944486727040t^{10} + 56631325373840304768t^{9} + 7536927788267588464t^{8} + 775768238916982984t^{7} + 61966116979388944t^{6} + 3688710659945624t^{5} + 143949145702218t^{4} + 2902950342163t^{3} + 28968626990t^{2} + 312509626t + 1168608 \right)D^{2} + 8 t ^{2} \left( 681472363461419335680t^{20} + 3221949619208664023040t^{19} + 6999408082664769519616t^{18} + 9826761091998139699200t^{17} + 10331369671331410912256t^{16} + 8559358139999122940416t^{15} + 5568820661468084210688t^{14} + 2794852021463681714432t^{13} + 1067986215484513315584t^{12} + 308866905179381227712t^{11} + 67536088047337656416t^{10} + 11206597062778283792t^{9} + 1425279837047526396t^{8} + 140820541978733938t^{7} + 10776013567915082t^{6} + 600964894964215t^{5} + 20917569730280t^{4} + 357116774250t^{3} + 3915290946t^{2} + 42355050t + 119196 \right)D + 48 t ^{2} \left( 46358664181048934400t^{20} + 215663943194694451200t^{19} + 460339199373014073344t^{18} + 637171452421172785152t^{17} + 663268471198980232192t^{16} + 544109301969721553408t^{15} + 348904556295228201984t^{14} + 171475173331316187904t^{13} + 63775196736326597376t^{12} + 17862035772047749696t^{11} + 3769440197077937632t^{10} + 602987962673535472t^{9} + 74070071674714572t^{8} + 7089301570989746t^{7} + 523395774352066t^{6} + 27541137141413t^{5} + 863251693498t^{4} + 12585957075t^{3} + 146756604t^{2} + 1323069t + 2772 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ3P3

|λ| = 18.87654 λ ≈ 18.87654; multiplicity 1; Jordan blocks 1 λ ≈ -3.718438 - 11.79456i; multiplicity 1; Jordan blocks 1 λ ≈ -3.718438 + 11.79456i; multiplicity 1; Jordan blocks 1 λ ≈ -2.438272 - 1.451602i; multiplicity 1; Jordan blocks 1 λ ≈ -2.438272 + 1.451602i; multiplicity 1; Jordan blocks 1 λ ≈ -2.563124; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 3 3
Sector dimension
9
Spectral radius
$18.876545$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.87654$11
$-3.718438 - 11.79456\mathrm{i}$11
$-3.718438 + 11.79456\mathrm{i}$11
$-2.438272 - 1.451602\mathrm{i}$11
$-2.438272 + 1.451602\mathrm{i}$11
$-2.563124$11
$-2$33

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{3} + 10 \lambda^{2} + 172 \lambda + 392\right) \left(\lambda^{3} - 14 \lambda^{2} - 84 \lambda - 152\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{3} + 10 \lambda^{2} + 172 \lambda + 392\right) \left(\lambda^{3} - 14 \lambda^{2} - 84 \lambda - 152\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 60 & 24 & 60 & 352 & 224 & 400 \\ 1 & 0 & 0 & 4 & 0 & 12 & 108 & 60 & 128 \\ 2 & 6 & 0 & 4 & 4 & 4 & 108 & 48 & 224 \\ 0 & 2 & 1 & 0 & 2 & 0 & 4 & 0 & 36 \\ 0 & 1 & 0 & 0 & -2 & 0 & 12 & 4 & 24 \\ 0 & 0 & 2 & 6 & 4 & 0 & 4 & 4 & 24 \\ 0 & 0 & 0 & 2 & 0 & 3 & 6 & 6 & 4 \\ 0 & 0 & 0 & 1 & 2 & -2 & -6 & -6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$