Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-180-43-a

·

This is a Fano fourfold of K3 type, appearing as case K3-50 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Fl(1, 2, 5) × ℙ4P4
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_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w4) + E^(w4 + w5) + E^(w1 + w2 + w5)
Bundle
wedge^2 Q_1,1^v ot U_1,2^v ot U_1,3^v + wedge^2 Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1) + O_1(1,1) box O_2(1)
ZL62 label
4042.7w6x4F00
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,4;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 43
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 180
$\chi(X, \mathrm{T}_X)$ = -18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 180
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 32
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 22 1 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 + 18 0 x0 + 18 x1
$\mathrm{HH}^{ 3 }$ = x1 + 31 0 0 x1 - 3 34
$\mathrm{HH}^{ 4 }$ = 44 0 0 1 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 38t + 102t^{2} + 38t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(130)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(172)

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( 20t^{2} + 4t + 1 \right) \left( 1783t^{4} + 836t^{3} - 36t^{2} + 9t - 1 \right) \left( 14754402214176t^{10} + 31493509985968t^{9} + 28468513217576t^{8} + 14712063970812t^{7} + 5101475925844t^{6} + 1237815976370t^{5} + 196086433225t^{4} + 16911667285t^{3} + 751662245t^{2} + 4994600t + 61200 \right)D^{6} + \left( 2t + 1 \right) ^{2} \left( 22097963284215678720t^{17} + 67693628758699307008t^{16} + 92846931834954049344t^{15} + 76336640991677277952t^{14} + 42796287918444705840t^{13} + 17571505920592884480t^{12} + 5457055759745254932t^{11} + 1285470101926389168t^{10} + 226399101092566012t^{9} + 29319908183922710t^{8} + 2737121915611595t^{7} + 183443864064452t^{6} + 11199911763810t^{5} + 1006388032695t^{4} + 77538657975t^{3} + 2971074780t^{2} + 14739000t + 122400 \right)D^{5} + \left( 2t + 1 \right) \left( 368299388070261312000t^{18} + 1241649683649204235520t^{17} + 1889464777516583846528t^{16} + 1733974295416884376640t^{15} + 1087499673276327379168t^{14} + 499912832473914340752t^{13} + 174550239921360482744t^{12} + 46654650962992434444t^{11} + 9433192890543894908t^{10} + 1413778387072405758t^{9} + 153572022943531119t^{8} + 11604491147265221t^{7} + 530449150193601t^{6} + 3559307587245t^{5} - 1463762314005t^{4} - 83712895240t^{3} - 2736920925t^{2} - 11587200t - 61200 \right)D^{4} + t \left( 3093714859790195020800t^{18} + 11509915540544374942720t^{17} + 19456596057431058826240t^{16} + 19944359844224358870784t^{15} + 14014944504291808935936t^{14} + 7232300871366514092096t^{13} + 2847248357497688226112t^{12} + 866578781999051194768t^{11} + 202573666375202831420t^{10} + 35703888345797349392t^{9} + 4642512335728828328t^{8} + 436098023948580532t^{7} + 28772815650751637t^{6} + 1270417463113478t^{5} + 34177094103700t^{4} + 359262852940t^{3} + 3285836925t^{2} + 28496590t + 68000 \right)D^{3} + 4 t ^{2} \left( 1708909160646012487680t^{17} + 6160579861251376704256t^{16} + 10080169308442703797248t^{15} + 10007137405851179761984t^{14} + 6828106071687943213088t^{13} + 3431981394560925542624t^{12} + 1316830033969383552968t^{11} + 388971919322850263232t^{10} + 87417383823536265016t^{9} + 14617297628558157808t^{8} + 1775760349192946464t^{7} + 153136084593073269t^{6} + 9051159506181207t^{5} + 348830637883087t^{4} + 8427507447675t^{3} + 117710107670t^{2} + 1665909700t + 7129800 \right)D^{2} + 4 t ^{2} \left( 1856228915874117012480t^{17} + 6534631753414321894912t^{16} + 10429867642756133029888t^{15} + 10104342359291774780928t^{14} + 6742953015787323341120t^{13} + 3322509890233445796032t^{12} + 1249431895016114366848t^{11} + 360024776545272723088t^{10} + 78227054033053168108t^{9} + 12499828875629859248t^{8} + 1432805683904541840t^{7} + 114930860112389264t^{6} + 6179333320828205t^{5} + 210707534238450t^{4} + 4421701066290t^{3} + 68803332580t^{2} + 954543200t + 3182400 \right)D + 64 t ^{2} \left( 47352778466176454400t^{17} + 163993393338468101760t^{16} + 257243165604598227456t^{15} + 244968732821902619664t^{14} + 160945725410507783456t^{13} + 78197954764543396376t^{12} + 28975829773177342678t^{11} + 8191830173256251368t^{10} + 1732951175584752682t^{9} + 266943145474954104t^{8} + 29183179457700650t^{7} + 2206142303379790t^{6} + 109811837207650t^{5} + 3418187569940t^{4} + 63002605175t^{3} + 1023401350t^{2} + 11874500t + 30600 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5) × ℙ4P4

|λ| = 12.437 λ ≈ 12.437; multiplicity 1; Jordan blocks 1 λ ≈ -0.7731134 - 8.673214i; multiplicity 1; Jordan blocks 1 λ ≈ -0.7731134 + 8.673214i; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 1, 2 λ ≈ -1.89077; multiplicity 1; Jordan blocks 1 2 2 4
Sector dimension
12
Spectral radius
$12.4369963$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.437$11
$-0.7731134 - 8.673214\mathrm{i}$11
$-0.7731134 + 8.673214\mathrm{i}$11
$-2 - 4\mathrm{i}$21, 1
$-2 + 4\mathrm{i}$21, 1
$-2$41, 1, 2
$-1.89077$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda^{2} + 4 \lambda + 20\right)^{2} \left(\lambda^{4} - 9 \lambda^{3} + 36 \lambda^{2} - 836 \lambda - 1783\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{4} - 9 \lambda^{3} + 36 \lambda^{2} - 836 \lambda - 1783\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 2 & 18 & 18 & 18 & 18 & 96 & 64 & 64 & 120 \\ 1 & -1 & 0 & 1 & 0 & 0 & 4 & 4 & 27 & 21 & 21 & 48 \\ 1 & 2 & 0 & 2 & 6 & -2 & -2 & -8 & -9 & -9 & -9 & 0 \\ 1 & 1 & 0 & -1 & 0 & 4 & 0 & 4 & 27 & 21 & 21 & 48 \\ 0 & 1 & 0 & 1 & -2 & 0 & 0 & 1 & 0 & 4 & 4 & 18 \\ 0 & 2 & 1 & 0 & 3 & -1 & 1 & -1 & 0 & 0 & -4 & 0 \\ 0 & 0 & 1 & 2 & 3 & 1 & -1 & -1 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 6 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 & 1 & -\frac{1}{3} & -2 & -\frac{2}{3} & -\frac{2}{3} & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{2} & 3 & 0 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & \frac{3}{2} & 3 & 2 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{3} & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$