Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-60-19-a

·

This is a Fano fourfold of K3 type, appearing as case K3-36 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
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_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(3w3) + E^(w1 + w2 + w3)
Bundle
O_3(3) + O_1(1) box O_2(1) box O_3(1)
ZL62 label
1140.Ex4000
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 19
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 60
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 60
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 132
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 5 5 0
0 1 22 1 0
0 5 5 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 + x2 + 23 0 x0 + 23 x2
$\mathrm{HH}^{ 3 }$ = x1 + 2 0 0 x1 2
$\mathrm{HH}^{ 4 }$ = x1 - x2 + x3 - 11 0 0 x1 - x2 - 30 x3 19
$\mathrm{HH}^{ 5 }$ = x3 - 1 0 0 x3 - 1 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 14t + 30t^{2} + 14t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(713)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(710)

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( 19t - 1 \right) \left( 4t + 1 \right) ^{2} \left( 8t + 1 \right) ^{4} \left( 97t^{2} - 46t + 1 \right) \left( 601392855874128t^{10} + 532226431161244t^{9} + 203504737218044t^{8} + 40759606512405t^{7} + 4597065124285t^{6} + 312973320891t^{5} + 14868105955t^{4} + 527600080t^{3} + 169824t^{2} - 1284340t + 1500 \right)D^{6} + \left( 4t + 1 \right) \left( 8t + 1 \right) ^{3} \left( 744822646428684031488t^{15} + 523616668729524621440t^{14} + 88417536090551713680t^{13} - 33914711845586148092t^{12} - 17792110414046305048t^{11} - 3353517503064815419t^{10} - 339331902110806121t^{9} - 22400001155181828t^{8} - 1159210486758438t^{7} - 29991106829063t^{6} + 2194174709207t^{5} + 179371289398t^{4} - 2960591372t^{3} + 51580196t^{2} - 3826020t + 3000 \right)D^{5} + \left( 8t + 1 \right) ^{2} \left( 198619372380982408396800t^{17} + 209327013723473597163520t^{16} + 85411050747145105773120t^{15} + 11833332392356723593904t^{14} - 2571026529292558970096t^{13} - 1368102705720227465112t^{12} - 264374522719810603324t^{11} - 30248682232668897591t^{10} - 2410161000589532548t^{9} - 148047265553717769t^{8} - 6347798262423020t^{7} - 54903727540597t^{6} + 10029475144216t^{5} + 132826315449t^{4} - 10070156340t^{3} - 53705920t^{2} + 2380280t - 1500 \right)D^{4} + t \left( 8t + 1 \right) \left( 6673610912001008922132480t^{17} + 7741229052980962767011840t^{16} + 3775118070159479402368000t^{15} + 893389022314459110840448t^{14} + 62076337669030450382560t^{13} - 20680272740682818798208t^{12} - 6700477686886266177948t^{11} - 1002499741440449895524t^{10} - 99228780294663138285t^{9} - 7340153689459451899t^{8} - 391220593942040848t^{7} - 8992401244621548t^{6} + 507344196615725t^{5} + 38142891744979t^{4} + 399068842616t^{3} - 6266409148t^{2} + 81045148t + 13500 \right)D^{3} + 2 t ^{2} \left( 58982008822256535997513728t^{17} + 74936036635691984491053056t^{16} + 41767469896095303982954496t^{15} + 12648235342074447616842752t^{14} + 2055233247970890500026784t^{13} + 101869291211821865999448t^{12} - 28323849269682208945440t^{11} - 7514098962836504395674t^{10} - 997523547818919937002t^{9} - 92854540107787718562t^{8} - 6352835689822084542t^{7} - 262341704685600285t^{6} + 555214111823711t^{5} + 696047339778177t^{4} + 28081881880247t^{3} + 224912486240t^{2} - 1125214130t + 21824220 \right)D^{2} + 8 t ^{2} \left( 16016666188802421413117952t^{17} + 20178978543984098094448640t^{16} + 11281977091452785209766912t^{15} + 3503211396446237689732352t^{14} + 629480504548725612343136t^{13} + 57806458654714192563080t^{12} - 334193486410787712392t^{11} - 879490464820235125442t^{10} - 143453763287690107372t^{9} - 14759254619089026457t^{8} - 1047972995272028179t^{7} - 38929208510091737t^{6} + 781113318770371t^{5} + 148293698453180t^{4} + 4961280481428t^{3} + 35845939476t^{2} + 27723380t + 1540800 \right)D + 16 t ^{2} \left( 3268707385469881921044480t^{17} + 4094715477969319423180800t^{16} + 2291619547444701892638720t^{15} + 720581433338613833027328t^{14} + 135810981485097179886864t^{13} + 14983069536607631914620t^{12} + 701963689932307635576t^{11} - 60505270891740594557t^{10} - 16441112741333113051t^{9} - 1957972700085378128t^{8} - 144293271943975300t^{7} - 4609317228519091t^{6} + 213886418696595t^{5} + 25102655690960t^{4} + 723844727624t^{3} + 4845622140t^{2} + 23506500t + 120000 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 43.78461 λ ≈ 43.78461; multiplicity 1; Jordan blocks 1 λ ≈ 19; multiplicity 2; Jordan blocks 1, 1 λ ≈ -8; multiplicity 6; Jordan blocks 1, 2, 3 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ 2.21539; multiplicity 1; Jordan blocks 1 2 6 2
Sector dimension
12
Spectral radius
$43.7846097$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$43.78461$11
$19$21, 1
$-8$61, 2, 3
$-4$22
$2.21539$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 8\right)^{6} \left(\lambda - 19\right)^{2} \left(\lambda^{2} - 46 \lambda + 97\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 8\right)^{3} \left(\lambda - 19\right) \left(\lambda^{2} - 46 \lambda + 97\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 14 & 14 & 132 & 36 & 360 & 360 & 1512 & 1296 & 4608 & 4608 & 15840 \\ 1 & -1 & 1 & 6 & 12 & 120 & 12 & 144 & 324 & 1188 & 432 & 3312 \\ 1 & 1 & -1 & 6 & 12 & 12 & 120 & 144 & 324 & 432 & 1188 & 3312 \\ 1 & 1 & 1 & 9 & 2 & 32 & 32 & 168 & 90 & 414 & 414 & 1296 \\ 0 & 1 & 1 & 0 & -2 & 6 & 6 & 0 & 108 & 144 & 144 & 864 \\ 0 & 1 & 0 & 1 & 1 & 8 & 1 & 15 & 30 & 138 & 30 & 324 \\ 0 & 0 & 1 & 1 & 1 & 1 & 8 & 15 & 30 & 30 & 138 & 324 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 2 & 14 & 14 & 36 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 7 & 13 & 13 & 104 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 1 & 0 & 2 & 14 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 2 & 0 & 14 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$