Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-120-31-e

·

This is a Fano fourfold of K3 type, appearing as case R-63 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ5P5 × ℙ7P7
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 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_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11node 12: fundamental weight ω_12ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12
Bundle
E^(2w6) + E^(w5 + w6) + 2 copies of E^(2w1)
Bundle
O_2(2) + Q_1 box O_2(1) + 2 copies of O_1(2)
ZL62 label
50700.Hw0cCC
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6;8) \\ \det\mathcal{E} &\cong \mathcal{O}(5;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 31
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 120
$\chi(X, \mathrm{T}_X)$ = -20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 120
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 3 0
0 2 2 0
0 1 22 1 0
0 2 2 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 + 20 0 x0 + 20 x2
$\mathrm{HH}^{ 3 }$ = x1 + 16 0 0 x1 16
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 19 0 0 x1 - x2 - 12 0 31
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 26t + 66t^{2} + 26t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(590)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(614)

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( 14t + 1 \right) \left( 18t - 1 \right) \left( 18t + 1 \right) \left( 2t - 1 \right) ^{2} \left( 2t + 1 \right) ^{3} \left( 1592349509376t^{9} + 1083991223040t^{8} + 318895536192t^{7} + 40632595776t^{6} + 1823652688t^{5} - 14534208t^{4} + 29154204t^{3} + 2788916t^{2} - 8868t + 135 \right)D^{6} + 2 \left( 2t - 1 \right) \left( 2t + 1 \right) ^{2} \left( 303361689730240512t^{14} + 204997036956383232t^{13} + 53260455001549824t^{12} + 1074684816757248t^{11} - 2195443802847744t^{10} - 438628520898816t^{9} - 30719552372096t^{8} - 1813021623744t^{7} - 714938315504t^{6} - 75372715176t^{5} - 1586597604t^{4} - 58061782t^{3} - 5803900t^{2} + 15732t - 135 \right)D^{5} + \left( 2t + 1 \right) \left( 20224112648682700800t^{16} + 12429936274372116480t^{15} + 688801961940295680t^{14} - 1655339765872914432t^{13} - 637971845044396032t^{12} - 83387915176372224t^{11} - 1037027234967552t^{10} + 270613047297536t^{9} - 156430638513024t^{8} - 16281342606784t^{7} + 2420697700640t^{6} + 319736918992t^{5} + 5296939024t^{4} - 17526352t^{3} - 11210260t^{2} + 4524t - 135 \right)D^{4} + 2 t \left( 84941273124467343360t^{16} + 90595810489620234240t^{15} + 32410487324222300160t^{14} + 608044040699154432t^{13} - 3145126558690148352t^{12} - 995771125032748032t^{11} - 127901451696877568t^{10} - 7325329316337664t^{9} - 927249118669696t^{8} - 237617446514816t^{7} - 7192598960864t^{6} + 2670765435536t^{5} + 239767844144t^{4} + 544866636t^{3} + 20258684t^{2} + 483974t + 81 \right)D^{3} + 4 t ^{2} \left( 93839882689887731712t^{15} + 96637756729326649344t^{14} + 37584927973250740224t^{13} + 4183171116680491008t^{12} - 1642539863856685056t^{11} - 629727534315373056t^{10} - 81905863162575104t^{9} - 4679061585288960t^{8} - 771321139513152t^{7} - 186692025606080t^{6} - 7927265108768t^{5} + 1059789880080t^{4} + 88651339444t^{3} - 187703470t^{2} + 18506496t + 169185 \right)D^{2} + 16 t ^{2} \left( 25482381937340203008t^{15} + 25555793773315940352t^{14} + 10407548164386428928t^{13} + 1628917475407566336t^{12} - 176506184981894400t^{11} - 99774228847479552t^{10} - 12580204667873088t^{9} - 538066054867584t^{8} - 142588479336016t^{7} - 38293852372320t^{6} - 2113598891812t^{5} + 119926727626t^{4} + 8543758734t^{3} - 33292487t^{2} + 2912298t + 15615 \right)D + 864 t ^{2} \left( 192610596654120960t^{15} + 189660574091427840t^{14} + 79218768636538880t^{13} + 14227040522916352t^{12} - 242931242338048t^{11} - 462346541707008t^{10} - 55542074930112t^{9} - 1163931783552t^{8} - 823471997872t^{7} - 239184872896t^{6} - 15419846308t^{5} + 483660782t^{4} + 24124002t^{3} - 52159t^{2} + 12657t + 45 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ5P5 × ℙ7P7

|λ| = 18 λ ≈ -18; multiplicity 1; Jordan blocks 1 λ ≈ 18; multiplicity 1; Jordan blocks 1 λ ≈ -14; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 3 λ ≈ 2; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$18$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-18$11
$18$11
$-14$11
$-2$33
$2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 14\right) \left(\lambda + 18\right) \left(\lambda - 18\right) \left(\lambda - 2\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 14\right) \left(\lambda + 18\right) \left(\lambda - 18\right) \left(\lambda - 2\right)^{2}\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 20 & 48 & 0 & 448 & 256 & 640 \\ 1 & -4 & 2 & 36 & 72 & 48 & -240 & 192 \\ 1 & 4 & 0 & 16 & -40 & 48 & 288 & 256 \\ 0 & 1 & 2 & 0 & 8 & 32 & -40 & 0 \\ 0 & 1 & 0 & 2 & -8 & 20 & 56 & 48 \\ 0 & 0 & 0 & 2 & 1 & 0 & 4 & 16 \\ 0 & 0 & 0 & 0 & 1 & 2 & -4 & 4 \\ 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0\end{pmatrix}\notag\end{equation}$$