Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-95-26-a

·

This is a Fano fourfold of K3 type, appearing as case K3-34 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Fl(1, 3, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1) + 2 copies of E^(w3) + E^(w1 + w3)
Bundle
O(1,0) + 2 copies of O(0,1) + O(1,1)
ZL62 label
44.220T
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 26
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 95
$\chi(X, \mathrm{T}_X)$ = -22

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 95
$\mathrm{c}_1^2\mathrm{c}_2$ = 110
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 22
$\mathrm{c}_4$ = 31
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 23 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 + 22 0 x0 + 22 x1
$\mathrm{HH}^{ 3 }$ = x1 + 19 0 0 x1 + 13 6
$\mathrm{HH}^{ 4 }$ = 27 0 0 1 0 26
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 21t + 51t^{2} + 21t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(604)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( 3t + 1 \right) ^{4} \left( 20389t^{6} + 24948t^{5} + 13857t^{4} + 3221t^{3} + 266t^{2} + 4t - 1 \right) \left( 246318029847720t^{13} + 726520321992816t^{12} + 928388496564738t^{11} + 684867089080882t^{10} + 328606040348530t^{9} + 110062955838217t^{8} + 27095640930126t^{7} + 5081044309079t^{6} + 731569619247t^{5} + 78126839044t^{4} + 5621506126t^{3} + 211845795t^{2} + 804089t + 7695 \right)D^{6} + \left( 3t + 1 \right) ^{3} \left( 316397233565605274040t^{20} + 1336261365213133992264t^{19} + 2596829234660492577930t^{18} + 3090226884707138181318t^{17} + 2525206758085614281568t^{16} + 1506451225669349285152t^{15} + 681822815738009276948t^{14} + 240644415672726476904t^{13} + 67661961220807215599t^{12} + 15401997310577258183t^{11} + 2863189528140053287t^{10} + 433605144742901506t^{9} + 52599365788814086t^{8} + 4944898881297091t^{7} + 343237511275282t^{6} + 16801169979031t^{5} + 621479772749t^{4} + 24576494734t^{3} + 842463113t^{2} + 2242977t + 15390 \right)D^{5} + \left( 3t + 1 \right) ^{2} \left( 7909930839140131851000t^{21} + 34251802189472292355440t^{20} + 68142832032448804661658t^{19} + 83013982684125259019598t^{18} + 69591497362534583266272t^{17} + 42778774507705886480005t^{16} + 20082012003209088583273t^{15} + 7412237714298115091254t^{14} + 2199610069951615087761t^{13} + 533418771613366733375t^{12} + 106599834869636293065t^{11} + 17510294789750108893t^{10} + 2326254764486086484t^{9} + 241999749392683148t^{8} + 18664510798922127t^{7} + 968473689831804t^{6} + 27033337509107t^{5} + 37046299123t^{4} - 21523177927t^{3} - 800251601t^{2} - 2414452t - 7695 \right)D^{4} + t \left( 3t + 1 \right) \left( 99665128573165661322600t^{21} + 447059799128102033577960t^{20} + 922241305260479601790830t^{19} + 1167013532449652376404646t^{18} + 1019095958174530860949410t^{17} + 655224853818000773343708t^{16} + 323395930895450172299082t^{15} + 126260570171006729112518t^{14} + 39892392270332796315863t^{13} + 10371547009999720944292t^{12} + 2239730382345480317842t^{11} + 401720317330097312633t^{10} + 59182070310064875223t^{9} + 6996616764602161670t^{8} + 639333099516539650t^{7} + 42617918308415056t^{6} + 1889774328808393t^{5} + 47784614259070t^{4} + 568143693536t^{3} + 6463922967t^{2} + 58958707t + 21384 \right)D^{3} + t ^{2} \left( 660637423684983812195520t^{21} + 3093640437579490773655056t^{20} + 6679164911724556855793388t^{19} + 8869965833648030059668348t^{18} + 8154589563991964358754506t^{17} + 5539755626109329709302406t^{16} + 2900611138533005336618199t^{15} + 1206483062479905986551795t^{14} + 407889466308494420408704t^{13} + 114007251922603361746470t^{12} + 26619973171976727681735t^{11} + 5204176050221611565611t^{10} + 845799362832026640480t^{9} + 112336002751121272314t^{8} + 11856811193274357443t^{7} + 953835464217492935t^{6} + 54915562104586072t^{5} + 2051013613748918t^{4} + 42994012967051t^{3} + 510487482855t^{2} + 7880127654t + 43537740 \right)D^{2} + 6 t ^{2} \left( 119598154287798793587120t^{21} + 548123645409020107749072t^{20} + 1154586725727313815449712t^{19} + 1492176902402217895432236t^{18} + 1332581049780209688131064t^{17} + 878526647054067158996292t^{16} + 446442338759820966914619t^{15} + 180437606011840288408615t^{14} + 59395691391458344249723t^{13} + 16197814676036116636773t^{12} + 3693844591262549390424t^{11} + 704464352969398402888t^{10} + 111248894514437092248t^{9} + 14259729813428432574t^{8} + 1438364676212136029t^{7} + 109081677670258949t^{6} + 5804776992708589t^{5} + 194690534997635t^{4} + 3575755693994t^{3} + 43403947254t^{2} + 655950246t + 2647080 \right)D + 36 t ^{2} \left( 8135928863115564189600t^{21} + 36738942545337255200160t^{20} + 76073928057634344142608t^{19} + 96460743405061193879808t^{18} + 84394015656377109022638t^{17} + 54463280871153628094022t^{16} + 27091576643183829148818t^{15} + 10726745131820296728550t^{14} + 3463944248595871398559t^{13} + 927926080539617535195t^{12} + 207922116193405804211t^{11} + 38902877559249012429t^{10} + 6005044560850799898t^{9} + 747961716121550308t^{8} + 72711219107128640t^{7} + 5253344848297280t^{6} + 261874438324461t^{5} + 8030025812013t^{4} + 133401268269t^{3} + 1712292507t^{2} + 21246786t + 61560 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 22.94787 λ ≈ 22.94787; multiplicity 1; Jordan blocks 1 λ ≈ -6.349585 - 6.031721i; multiplicity 1; Jordan blocks 1 λ ≈ -6.349585 + 6.031721i; multiplicity 1; Jordan blocks 1 λ ≈ -3.595643; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 4; Jordan blocks 2, 2 λ ≈ -1.32653 - 1.209148i; multiplicity 1; Jordan blocks 1 λ ≈ -1.32653 + 1.209148i; multiplicity 1; Jordan blocks 1 4
Sector dimension
10
Spectral radius
$22.947874$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22.94787$11
$-6.349585 - 6.031721\mathrm{i}$11
$-6.349585 + 6.031721\mathrm{i}$11
$-3.595643$11
$-3$42, 2
$-1.32653 - 1.209148\mathrm{i}$11
$-1.32653 + 1.209148\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{4} \left(\lambda^{6} - 4 \lambda^{5} - 266 \lambda^{4} - 3221 \lambda^{3} - 13857 \lambda^{2} - 24948 \lambda - 20389\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda^{6} - 4 \lambda^{5} - 266 \lambda^{4} - 3221 \lambda^{3} - 13857 \lambda^{2} - 24948 \lambda - 20389\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 20 & 174 & 78 & 120 & 81 & 224 & 536 & 890 \\ 1 & -1 & 5 & 16 & 8 & 12 & 8 & 33 & 87 & 144 \\ 1 & 2 & 0 & 34 & 18 & 18 & 12 & 45 & 105 & 160 \\ 0 & 1 & 1 & -1 & 0 & 4 & 3 & 2 & 8 & 24 \\ 0 & 1 & 0 & 5 & -1 & 2 & 1 & 4 & 6 & 12 \\ 0 & 0 & 1 & 2 & 1 & -2 & 0 & 6 & 14 & 18 \\ 0 & 0 & 1 & 2 & 2 & 0 & -2 & 2 & 8 & 12 \\ 0 & 0 & 0 & -3 & -\frac{19}{5} & \frac{13}{5} & 2 & -6 & -5 & \frac{8}{5} \\ 0 & 0 & 0 & 5 & \frac{16}{5} & \frac{8}{5} & 1 & 4 & 5 & \frac{48}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$