Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-261-58-a

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
Fl(1, 2, 6) × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 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_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(w2 + w6) + E^(w5) + E^(w5 + w6) + E^(w1 + w2)
Bundle
O_1(0,1) box O_2(1) + wedge^3 Q_1,1^v ot U_1,2^v ot U_1,3^v + wedge^3 Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1) + O_1(1,1)
ZL62 label
5052.9wZw8w5
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1,4;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 58
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 261
$\chi(X, \mathrm{T}_X)$ = -2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 261
$\mathrm{c}_1^2\mathrm{c}_2$ = 162
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 50
$\mathrm{c}_4$ = 17
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 9 0 0
0 0 0 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 + x1 + 2 0 x0 + 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 44 0 0 x1 - 18 62
$\mathrm{HH}^{ 4 }$ = 58 0 0 0 0 58
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 53t + 153t^{2} + 53t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(30)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(35)

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) \left( 2t + 1 \right) \left( 17t^{2} + 2t + 1 \right) \left( 19t^{2} + 7t - 1 \right) \left( 119t^{3} + 65t^{2} + 6t + 1 \right) \left( 7342012208938t^{13} + 169761118213396t^{12} + 571232416793756t^{11} + 858475646808877t^{10} + 693731974822636t^{9} + 320250866909815t^{8} + 85072524492392t^{7} + 13108667926339t^{6} + 1512248542830t^{5} + 223818011981t^{4} + 24946365904t^{3} + 992603724t^{2} + 8436712t + 182700 \right)D^{6} + 2 \left( 5926303388773948026t^{22} + 155700981887659981135t^{21} + 811276310913246487549t^{20} + 2071741343097022293849t^{19} + 3207604629828387093985t^{18} + 3299808622838043966709t^{17} + 2367620021020671278365t^{16} + 1219328401032846329799t^{15} + 459753123559437404785t^{14} + 129451001602886886445t^{13} + 28068076847334335213t^{12} + 4943906812417791804t^{11} + 760928819817833540t^{10} + 108089042227244302t^{9} + 14025531427250050t^{8} + 1515864626639348t^{7} + 118180479671488t^{6} + 7149612037277t^{5} + 694309127699t^{4} + 62225112040t^{3} + 1955179624t^{2} + 11924268t + 182700 \right)D^{5} + \left( 98771723146232467100t^{22} + 2643904539719848149050t^{21} + 13334622031627430364346t^{20} + 32720199931755443317464t^{19} + 48361410262741244004983t^{18} + 47155425361635322736179t^{17} + 31806414770865772954950t^{16} + 15251093728446261857532t^{15} + 5297450840688882626151t^{14} + 1363076768602551380440t^{13} + 271592431274874990897t^{12} + 45455342390525461500t^{11} + 6900406614232054341t^{10} + 925349489536846567t^{9} + 96693534435737600t^{8} + 6769795656164408t^{7} + 251132648809029t^{6} - 2458112109684t^{5} - 1372951203277t^{4} - 124637489168t^{3} - 3970892604t^{2} - 23245104t - 182700 \right)D^{4} + 2 t \left( 207420618607088180910t^{21} + 5632865022038781077725t^{20} + 27646540006871469254095t^{19} + 65578606782901788794151t^{18} + 93104179457829318950625t^{17} + 86628373938858908591295t^{16} + 55369479705315595262078t^{15} + 24973911157754154842560t^{14} + 8107317566189691837681t^{13} + 1949108343859252693265t^{12} + 369567747129882819101t^{11} + 61449952735273139893t^{10} + 9534551961615543953t^{9} + 1279805821047938179t^{8} + 131500837008217774t^{7} + 9585897885061052t^{6} + 473547470858739t^{5} + 14640230852632t^{4} + 214284023784t^{3} + 3940481016t^{2} + 13056092t + 43400 \right)D^{3} + 4 t ^{2} \left( 229150397699259323672t^{20} + 6291156234972892637000t^{19} + 30211918589058064101916t^{18} + 69710127863110943218830t^{17} + 95701415022521999484027t^{16} + 85592680523102998032419t^{15} + 52272141516246393056986t^{14} + 22397353264033709304653t^{13} + 6882068643004326772884t^{12} + 1574052989801762370943t^{11} + 291263653089618738155t^{10} + 49062452799508555875t^{9} + 7681607529454135556t^{8} + 987529644828766482t^{7} + 92162593977373003t^{6} + 5916464527055618t^{5} + 260113527534841t^{4} + 8093238930052t^{3} + 148966307396t^{2} + 2597078304t + 4517388 \right)D^{2} + 8 t ^{2} \left( 124452371164252908546t^{20} + 3443884742813007254935t^{19} + 16268245323900086873389t^{18} + 36745558575574331263653t^{17} + 49127794868308392529655t^{16} + 42570142697748039865971t^{15} + 25060169440238110554050t^{14} + 10302252076385449821636t^{13} + 3031209387450851091923t^{12} + 668942613413934986476t^{11} + 122635190892104545421t^{10} + 21019737135106388305t^{9} + 3288365457617005598t^{8} + 403098630592815844t^{7} + 34458493616872650t^{6} + 1980763418040044t^{5} + 78861061402472t^{4} + 2427618181184t^{3} + 49391534072t^{2} + 761135536t + 584640 \right)D + 96 t ^{3} \left( 4233073849124248590t^{19} + 117762510490138620025t^{18} + 550010173020906643339t^{17} + 1223907436385117464743t^{16} + 1605597511946098479297t^{15} + 1359565533309019488598t^{14} + 778890951810624644999t^{13} + 310453339078858408185t^{12} + 88452751723865447304t^{11} + 19046218958195001753t^{10} + 3485431186110413249t^{9} + 606236906757079561t^{8} + 94237582434800429t^{7} + 11071563068067240t^{6} + 880603752242833t^{5} + 46317825825091t^{4} + 1711809980708t^{3} + 51845520600t^{2} + 1113048756t + 14798700 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6) × ℙ5P5

|λ| = 9.09017 λ ≈ 9.09017; multiplicity 1; Jordan blocks 1 λ ≈ -1.952838 - 7.280568i; multiplicity 1; Jordan blocks 1 λ ≈ -1.952838 + 7.280568i; multiplicity 1; Jordan blocks 1 λ ≈ -1 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.094324; multiplicity 1; Jordan blocks 1 λ ≈ -2.09017; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 1, 1 2 2 2
Sector dimension
12
Spectral radius
$9.09016994$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.09017$11
$-1.952838 - 7.280568\mathrm{i}$11
$-1.952838 + 7.280568\mathrm{i}$11
$-1 - 4\mathrm{i}$21, 1
$-1 + 4\mathrm{i}$21, 1
$-2.094324$11
$-2.09017$11
$-2$11
$-1$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{2} + 2 \lambda + 17\right)^{2} \left(\lambda^{2} - 7 \lambda - 19\right) \left(\lambda^{3} + 6 \lambda^{2} + 65 \lambda + 119\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda^{2} + 2 \lambda + 17\right) \left(\lambda^{2} - 7 \lambda - 19\right) \left(\lambda^{3} + 6 \lambda^{2} + 65 \lambda + 119\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 12 & 12 & 12 & 12 & 32 & 24 & 24 & 20 \\ 1 & -1 & 0 & 0 & 0 & 0 & 4 & 4 & 12 & 12 & 12 & 16 \\ 1 & 2 & 0 & 2 & 0 & -4 & -4 & -8 & 0 & -6 & -6 & 0 \\ 1 & 0 & 0 & -1 & 0 & 4 & 0 & 4 & 12 & 12 & 12 & 16 \\ 0 & 1 & 0 & 1 & -2 & 0 & 0 & 1 & 0 & 4 & 4 & 12 \\ 0 & 2 & 1 & 0 & 2 & -1 & 0 & -1 & 0 & 0 & -4 & 0 \\ 0 & 0 & 1 & 2 & 2 & 0 & -1 & -1 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 & 1 & -\frac{1}{4} & -2 & -\frac{1}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{2} & 2 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & \frac{3}{2} & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$