Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-114-30-b

·

This is a Fano fourfold of K3 type, appearing as case GM-21 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Fl(2, 3, 5)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1) + E^(w2) + E^(2w3)
Bundle
U_3^v + O(1,0) + O(0,2)
ZL62 label
45.610
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 114
$\chi(X, \mathrm{T}_X)$ = -24

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 114
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 122
$\mathrm{c}_1\mathrm{c}_3$ = 20
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 24 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 + x2 + 24 0 x0 + 24 x2
$\mathrm{HH}^{ 3 }$ = x1 + 14 0 0 x1 14
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 21 0 0 x1 - x2 - 9 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 62t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(558)
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) \left( 2t + 1 \right) ^{2} \left( 7504t^{6} + 5984t^{5} - 1576t^{4} - 2112t^{3} - 215t^{2} - 2t + 1 \right) \left( 150679404995584t^{13} + 241904894110720t^{12} + 81468944409344t^{11} - 126990358638336t^{10} - 164421464649216t^{9} - 87205793451584t^{8} - 24743080383840t^{7} - 3720000720128t^{6} - 229748313316t^{5} + 6834612448t^{4} + 1515395223t^{3} - 92347031t^{2} - 2378491t - 4161 \right)D^{6} + \left( 2t + 1 \right) \left( 142467980140944654336t^{21} + 414124430643388825600t^{20} + 421733366481086046208t^{19} + 18668620112202493952t^{18} - 376040458694760605696t^{17} - 408813430941458756608t^{16} - 205413371958565118464t^{15} - 37250157965663510784t^{14} + 16045673367378889600t^{13} + 13085175168830013120t^{12} + 4263672732540115360t^{11} + 791875730462118096t^{10} + 82879728905462568t^{9} + 3543657040319476t^{8} - 88508203464598t^{7} + 8089422190053t^{6} + 3633982234915t^{5} + 120284724024t^{4} - 5589865608t^{3} + 348658989t^{2} + 7077219t + 8322 \right)D^{5} + \left( 2374466335682410905600t^{22} + 7463563123388561489920t^{21} + 8727401157044937261056t^{20} + 2045725732862585995264t^{19} - 6463158324810212696064t^{18} - 9117423755348253986816t^{17} - 6135978376435375122432t^{16} - 2356886719877574526976t^{15} - 421014642045078577664t^{14} + 52656019975989992448t^{13} + 52211030801417535872t^{12} + 14314895861159260928t^{11} + 2031516206154517024t^{10} + 117829393316906928t^{9} - 7962165805163544t^{8} - 1605135430904904t^{7} - 24645872849963t^{6} + 7493020768774t^{5} + 347775568809t^{4} + 13359537168t^{3} - 325312037t^{2} - 5334558t - 4161 \right)D^{4} + t \left( 9972758609866125803520t^{21} + 29282010286151847772160t^{20} + 31165558648812610846720t^{19} + 3100798516035112648704t^{18} - 28519519359518388772864t^{17} - 36214172424161295298560t^{16} - 23702960201072494671872t^{15} - 9438602688028973036544t^{14} - 2199097027701966839296t^{13} - 187776376232243783936t^{12} + 49832070223359175424t^{11} + 19025968191701062336t^{10} + 2683951694623623136t^{9} + 129972600923150928t^{8} - 8434289546069448t^{7} - 25217485612244t^{6} + 283758961361829t^{5} + 24789600669293t^{4} + 617936352228t^{3} + 7072316480t^{2} + 12452759t + 7519 \right)D^{3} + 2 t ^{2} \left( 11017523797566386601984t^{20} + 30604270969966709866496t^{19} + 29972725579154737987584t^{18} - 925636170077877133312t^{17} - 32560308631892353388544t^{16} - 38279761263865699756032t^{15} - 24314326643847818478592t^{14} - 9696023806115538134016t^{13} - 2433697065623637778176t^{12} - 335190967841085463424t^{11} - 4350535852788909696t^{10} + 7139207151228083488t^{9} + 1223304505864502672t^{8} + 71082495063820776t^{7} + 842249323014940t^{6} + 1152931625506424t^{5} + 237401278793771t^{4} + 13207516591953t^{3} + 265550332667t^{2} + 2573989411t + 7314810 \right)D^{2} + 8 t ^{2} \left( 2991827582959837741056t^{20} + 7963314232177328570368t^{19} + 7277178663179033509888t^{18} - 1072497191232700665856t^{17} - 9018959319650399127552t^{16} - 10032611680068128438272t^{15} - 6203393666929469894144t^{14} - 2447999835112398843136t^{13} - 626629350089537798144t^{12} - 97124276167928973312t^{11} - 6197832563871580384t^{10} + 700028821740603968t^{9} + 187068355067933312t^{8} + 15810298014726524t^{7} + 1324344452745036t^{6} + 354098502150998t^{5} + 45083084952073t^{4} + 1864052062201t^{3} + 33868772175t^{2} + 245498921t + 832200 \right)D + 32 t ^{2} \left( 305288528873452830720t^{20} + 788638803468000706560t^{19} + 684202625728650412032t^{18} - 164058593902250188800t^{17} - 930802313876254623744t^{16} - 997589278809100658688t^{15} - 604257575544289382400t^{14} - 235620391198619499264t^{13} - 60426734451316714496t^{12} - 9730994739497707776t^{11} - 778305862836082464t^{10} + 27623037475402864t^{9} + 13929766696489040t^{8} + 1532497592471540t^{7} + 174400406689098t^{6} + 34042535942102t^{5} + 3336505401116t^{4} + 111424791769t^{3} + 1924132199t^{2} + 9731849t + 29127 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5)

|λ| = 19.156 λ ≈ 19.156; multiplicity 1; Jordan blocks 1 λ ≈ -8.002105 - 5.823893i; multiplicity 1; Jordan blocks 1 λ ≈ -8.002105 + 5.823893i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ 1.741879; multiplicity 1; Jordan blocks 1 λ ≈ -1.446836 - 0.4501075i; multiplicity 1; Jordan blocks 1 λ ≈ -1.446836 + 0.4501075i; multiplicity 1; Jordan blocks 1 2
Sector dimension
9
Spectral radius
$19.1560034$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.156$11
$-8.002105 - 5.823893\mathrm{i}$11
$-8.002105 + 5.823893\mathrm{i}$11
$-3$11
$-2$22
$1.741879$11
$-1.446836 - 0.4501075\mathrm{i}$11
$-1.446836 + 0.4501075\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right) \left(\lambda^{6} - 2 \lambda^{5} - 215 \lambda^{4} - 2112 \lambda^{3} - 1576 \lambda^{2} + 5984 \lambda + 7504\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right) \left(\lambda^{6} - 2 \lambda^{5} - 215 \lambda^{4} - 2112 \lambda^{3} - 1576 \lambda^{2} + 5984 \lambda + 7504\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 16 & 24 & 84 & 60 & 112 & 48 & 320 \\ 1 & -1 & 4 & 4 & 4 & 4 & 36 & 24 & 64 \\ 1 & 1 & 0 & 8 & 20 & 12 & 24 & 0 & 48 \\ 0 & 3 & 1 & -1 & 8 & 6 & 4 & 4 & 36 \\ 0 & 1 & 2 & 0 & 0 & 2 & 8 & 8 & 24 \\ 0 & 0 & 1 & 1 & 0 & -2 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 5 & 4 & 0 & 1 & 12 \\ 0 & 0 & 0 & 2 & 1 & 0 & 4 & -1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0\end{pmatrix}\notag\end{equation}$$