Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-160-39-a

·

This is a Fano fourfold of K3 type, appearing as case K3-29 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ2P2 × ℙ4P4
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + w3) + E^(w1 + 2w3)
Bundle
O_1(1) box O_2(1) + O_1(1) box O_2(2)
ZL62 label
2040.hx311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 160
$\chi(X, \mathrm{T}_X)$ = -21

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 160
$\mathrm{c}_1^2\mathrm{c}_2$ = 136
$\mathrm{c}_2^2$ = 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 + 21 0 x0 + 21 x1
$\mathrm{HH}^{ 3 }$ = x1 + 32 0 0 x1 + 1 31
$\mathrm{HH}^{ 4 }$ = 40 0 0 1 0 39
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 90t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(254)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(322)

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 + 1 \right) ^{3} \left( 12469t^{6} + 11852t^{5} + 5032t^{4} + 1472t^{3} + 48t^{2} - t - 1 \right) \left( 434175435972t^{10} + 1261336864212t^{9} + 1682867152362t^{8} + 1553688078309t^{7} + 1087687372620t^{6} + 463273757144t^{5} + 92093741952t^{4} + 7276107072t^{3} + 1438106176t^{2} + 10750720t + 128000 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 113688403733832228t^{17} + 498451611591908280t^{16} + 1025740703422810194t^{15} + 1375577334580531116t^{14} + 1374822161846494515t^{13} + 1045747463280004144t^{12} + 585254193682513076t^{11} + 233562851858849542t^{10} + 65746248124677118t^{9} + 13133392879322145t^{8} + 1903671034042721t^{7} + 222783251986360t^{6} + 23003131094888t^{5} + 956154831104t^{4} + 33638990080t^{3} + 5712625664t^{2} + 31100160t + 256000 \right)D^{5} + \left( t + 1 \right) \left( 947403364448601900t^{18} + 4743970918988404500t^{17} + 11075083171688972538t^{16} + 16602247183329002127t^{15} + 18359328391616259474t^{14} + 15600514133324151795t^{13} + 9962618381410463668t^{12} + 4604539668763935492t^{11} + 1503425407806579071t^{10} + 348012444686714713t^{9} + 59689311002989259t^{8} + 8124694228844609t^{7} + 883563315210512t^{6} + 58728632098400t^{5} + 210196090576t^{4} - 14566999328t^{3} - 5459577280t^{2} - 31171840t - 128000 \right)D^{4} + t \left( 3979094130684127980t^{18} + 22725025601639297760t^{17} + 60271721120623825650t^{16} + 101501531021560559556t^{15} + 124726495358897133579t^{14} + 118257748716621207896t^{13} + 86072265638724696757t^{12} + 46541367383694894004t^{11} + 18119543525099495030t^{10} + 5011117000071320355t^{9} + 1008128338308740505t^{8} + 158854784349469991t^{7} + 20633281732293483t^{6} + 2024747514076496t^{5} + 110840990958408t^{4} + 1657670930720t^{3} + 24469935488t^{2} + 186052352t + 96000 \right)D^{3} + 2 t ^{2} \left( 4395951611041512816t^{17} + 24109421216995257912t^{16} + 61606545968753565852t^{15} + 100767790681598038878t^{14} + 121383697313786937969t^{13} + 112958481585841258334t^{12} + 79848897177806801132t^{11} + 41213887193060849674t^{10} + 15045810786040497067t^{9} + 3859877351456290947t^{8} + 728421785684256748t^{7} + 111391773237485894t^{6} + 13886116250236908t^{5} + 1205484474940026t^{4} + 52461052477100t^{3} + 791517842232t^{2} + 14079301840t + 72412160 \right)D^{2} + 8 t ^{2} \left( 1193728239205238394t^{17} + 6348688094038783428t^{16} + 15770804723904677841t^{15} + 25246594289933648874t^{14} + 29984281036741071876t^{13} + 27502106941342818570t^{12} + 18965401938670987613t^{11} + 9405018363404273002t^{10} + 3250802157641026742t^{9} + 784905930883767586t^{8} + 141828237247748298t^{7} + 21293524002413160t^{6} + 2521024388677152t^{5} + 192131688283584t^{4} + 6719253289216t^{3} + 112393218048t^{2} + 1912133120t + 6963200 \right)D + 32 t ^{2} \left( 121809004000534530t^{17} + 634157106200662860t^{16} + 1544542682430360681t^{15} + 2436193500733349028t^{14} + 2865844637982032619t^{13} + 2601117065303230368t^{12} + 1759728895227270927t^{11} + 845577314605529960t^{10} + 279851686801894059t^{9} + 64473111496763463t^{8} + 11319174241041835t^{7} + 1675501216953110t^{6} + 188050912485196t^{5} + 12731311999112t^{4} + 367804975344t^{3} + 7033927040t^{2} + 103916800t + 256000 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 13.35719 λ ≈ 13.35719; multiplicity 1; Jordan blocks 1 λ ≈ -5.400287 - 8.207351i; multiplicity 1; Jordan blocks 1 λ ≈ -5.400287 + 8.207351i; multiplicity 1; Jordan blocks 1 λ ≈ -0.9266374 - 2.195255i; multiplicity 1; Jordan blocks 1 λ ≈ -0.9266374 + 2.195255i; multiplicity 1; Jordan blocks 1 λ ≈ -1.703344; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
9
Spectral radius
$13.3571929$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.35719$11
$-5.400287 - 8.207351\mathrm{i}$11
$-5.400287 + 8.207351\mathrm{i}$11
$-0.9266374 - 2.195255\mathrm{i}$11
$-0.9266374 + 2.195255\mathrm{i}$11
$-1.703344$11
$-1$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{6} + \lambda^{5} - 48 \lambda^{4} - 1472 \lambda^{3} - 5032 \lambda^{2} - 11852 \lambda - 12469\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{6} + \lambda^{5} - 48 \lambda^{4} - 1472 \lambda^{3} - 5032 \lambda^{2} - 11852 \lambda - 12469\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 30 & 12 & 30 & 56 & 32 & 75 \\ 1 & -1 & 0 & 4 & 0 & 10 & 33 & 18 & 12 \\ 2 & 3 & 0 & 0 & 0 & 4 & 54 & 24 & 48 \\ 0 & 2 & 1 & -1 & 0 & 0 & 4 & 0 & 18 \\ 0 & 1 & 0 & 0 & -1 & 0 & 10 & 4 & 9 \\ 0 & 0 & 2 & 3 & 2 & 0 & 0 & 0 & 18 \\ 0 & 0 & 0 & 2 & 0 & 4 & \frac{7}{2} & 3 & 0 \\ 0 & 0 & 0 & 1 & 2 & -\frac{7}{2} & -\frac{21}{4} & -\frac{9}{2} & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$