Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-192-45-e

·

This is a Fano fourfold of K3 type, appearing as case C-3 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ2P2 × ℙ5P5
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_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w1 + 2w3) + E^(w2 + w3)
Bundle
O_1(1) box O_2(2) + Q_1 box O_2(1)
ZL62 label
2050.w6x311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 192
$\chi(X, \mathrm{T}_X)$ = -19

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 192
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 114
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 30
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 22 1 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 19 0 x0 + 19 x1
$\mathrm{HH}^{ 3 }$ = x1 + 36 0 0 x1 - 9 45
$\mathrm{HH}^{ 4 }$ = 46 0 0 1 0 45
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(246)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(314)

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( t^{2} + 11t - 1 \right) \left( 1369t^{4} + 265t^{3} + 110t^{2} + 14t + 1 \right) \left( 2159675200t^{10} + 1354069212t^{9} - 879851795t^{8} + 663097599t^{7} + 6038657020t^{6} + 2444690448t^{5} + 2206882640t^{4} - 128849104t^{3} + 6054912t^{2} + 4160t + 512 \right)D^{6} + \left( 35479144185600t^{16} + 288857271884364t^{15} + 184190258464022t^{14} - 110000086794975t^{13} + 236212730447104t^{12} + 1173199947376042t^{11} + 623435073752541t^{10} + 529673452813675t^{9} + 18817943607715t^{8} + 12024417404535t^{7} - 474610782400t^{6} + 59643567264t^{5} + 13992407856t^{4} - 638399440t^{3} + 24345408t^{2} + 9408t + 1024 \right)D^{5} + \left( 171482530230400t^{16} + 978529108879836t^{15} + 540762975661329t^{14} - 490117100473003t^{13} + 943424324271104t^{12} + 5452058185257999t^{11} + 3021842915225666t^{10} + 2717065619847848t^{9} - 13248734583516t^{8} + 26798310274219t^{7} - 1821550566592t^{6} + 121647676920t^{5} - 19536551632t^{4} + 694748400t^{3} - 24161088t^{2} - 73408t - 512 \right)D^{4} + t \left( 425749730227200t^{15} + 1743755413092660t^{14} + 745081966933588t^{13} - 1109105985433029t^{12} + 1617528558615984t^{11} + 12587968341929772t^{10} + 7575223668767403t^{9} + 7226651936504287t^{8} - 211419542573921t^{7} + 34865482126541t^{6} - 1348890537160t^{5} + 296439588560t^{4} + 2322451696t^{3} + 57570320t^{2} + 647616t + 64 \right)D^{3} + 2 t ^{2} \left( 285311451159200t^{14} + 864710631636156t^{13} + 235665763140802t^{12} - 740376212862050t^{11} + 630246212007938t^{10} + 7758569305553784t^{9} + 5212952456085732t^{8} + 5133536353856551t^{7} - 224081280782162t^{6} + 18011994637270t^{5} + 600612037906t^{4} + 151228313732t^{3} + 781140616t^{2} + 36279264t + 245760 \right)D^{2} + 8 t ^{2} \left( 48783823255200t^{14} + 113387233057542t^{13} + 11030966482356t^{12} - 136751671486443t^{11} + 47125960577170t^{10} + 1223817772478162t^{9} + 925169059340642t^{8} + 916167038823990t^{7} - 47999983439240t^{6} + 3381749771872t^{5} + 275225538304t^{4} + 19131043360t^{3} + 112317952t^{2} + 5568128t + 24576 \right)D + 32 t ^{2} \left( 3326169767400t^{14} + 6150362305287t^{13} - 553006447344t^{12} - 10437128962494t^{11} + 231475752163t^{10} + 77692808155242t^{9} + 65665360186687t^{8} + 64350391427595t^{7} - 3724929357466t^{6} + 282863064332t^{5} + 26369554648t^{4} + 969424336t^{3} + 8462720t^{2} + 341312t + 1024 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ5P5

|λ| = 11.09017 λ ≈ 11.09017; multiplicity 1; Jordan blocks 1 λ ≈ -6.699066 - 6.414356i; multiplicity 1; Jordan blocks 1 λ ≈ -6.699066 + 6.414356i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3009344 - 3.977948i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3009344 + 3.977948i; multiplicity 1; Jordan blocks 1 λ ≈ -0.09016994; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
9
Spectral radius
$11.0901699$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.09017$11
$-6.699066 - 6.414356\mathrm{i}$11
$-6.699066 + 6.414356\mathrm{i}$11
$-0.3009344 - 3.977948\mathrm{i}$11
$-0.3009344 + 3.977948\mathrm{i}$11
$-0.09016994$11
$0$31, 2

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda^{2} - 11 \lambda - 1\right) \left(\lambda^{4} + 14 \lambda^{3} + 110 \lambda^{2} + 265 \lambda + 1369\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda^{2} - 11 \lambda - 1\right) \left(\lambda^{4} + 14 \lambda^{3} + 110 \lambda^{2} + 265 \lambda + 1369\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 18 & 12 & 18 & 0 & 0 & 60 \\ 1 & -1 & 0 & 4 & 0 & 10 & 3 & 6 & 0 \\ 2 & 1 & 0 & 0 & 0 & 4 & 42 & 24 & 0 \\ 0 & 2 & 1 & -1 & -2 & 0 & 4 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 & 4 & 0 \\ 0 & 0 & 2 & 1 & 2 & 0 & 0 & 0 & 18 \\ 0 & 0 & 0 & 2 & 0 & 4 & \frac{1}{2} & 1 & 0 \\ 0 & 0 & 0 & 1 & 2 & -\frac{3}{2} & -\frac{3}{4} & -\frac{3}{2} & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$