Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-240-54-g

·

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

Defining data
Ambient
ℙ1P1 × Gr(2, 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_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w3) + E^(w1 + w3)
Bundle
2 copies of O_2(1) + O_1(1) box O_2(1)
ZL62 label
141.22a
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 240
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 240
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 101
$\mathrm{c}_1\mathrm{c}_3$ = 46
$\mathrm{c}_4$ = 13
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 7 0 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 + 14 0 x0 + 4 10
$\mathrm{HH}^{ 3 }$ = 56 0 0 0 56
$\mathrm{HH}^{ 4 }$ = 54 0 0 0 0 54
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 49t + 140t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(288)
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( t + 1 \right) \left( 11t^{2} + 9t - 1 \right) \left( 389t^{6} + 81t^{5} - 60t^{4} + 156t^{3} - 23t^{2} - 11t - 1 \right) \left( 14821186560t^{13} + 84694384800t^{12} + 191811373536t^{11} + 216509277056t^{10} + 135704642340t^{9} + 56176006640t^{8} + 20000445535t^{7} + 3153564970t^{6} - 1437358146t^{5} - 228602952t^{4} + 23622284t^{3} + 1338712t^{2} + 15852t + 432 \right)D^{6} + \left( 1331817003095040t^{22} + 10157706643552320t^{21} + 32881004471847072t^{20} + 58767003960159840t^{19} + 64607151400188700t^{18} + 47884210359928332t^{17} + 27679266810314387t^{16} + 14369327977084139t^{15} + 6233062873201258t^{14} + 2015864209888485t^{13} + 609920904435690t^{12} + 118106955880409t^{11} - 55556120449411t^{10} - 24308759001655t^{9} + 2419518245685t^{8} + 926126617611t^{7} - 55540344562t^{6} - 538045394t^{5} + 768522248t^{4} - 127315240t^{3} - 5527612t^{2} - 44964t - 864 \right)D^{5} + \left( 11098475025792000t^{22} + 83636550282350400t^{21} + 266575850648083200t^{20} + 467369303587393184t^{19} + 502406538738797156t^{18} + 359775223803741316t^{17} + 192456912640311413t^{16} + 85207047423800371t^{15} + 28398457846310632t^{14} + 6000495178327588t^{13} + 1672567883239374t^{12} + 581375468607309t^{11} - 82864244255376t^{10} - 55215135354750t^{9} + 6763262640509t^{8} + 3050986787525t^{7} - 95667083996t^{6} - 39893510310t^{5} - 3716615236t^{4} + 91081124t^{3} + 5267732t^{2} + 63636t + 432 \right)D^{4} + t \left( 46613595108326400t^{21} + 347937194419939200t^{20} + 1093917428182554720t^{19} + 1884604055731977120t^{18} + 1989061384136761108t^{17} + 1398294175873865988t^{16} + 726029247687428693t^{15} + 294535718759843821t^{14} + 73912512441573894t^{13} + 1401030563198783t^{12} - 2085293704791540t^{11} + 305247851685955t^{10} - 240514485599912t^{9} - 132552757369777t^{8} - 1610193447325t^{7} + 4612170467545t^{6} - 59765153466t^{5} - 20666738758t^{4} - 1119950940t^{3} - 42700528t^{2} - 718196t - 108 \right)D^{3} + t ^{2} \left( 102993848239349760t^{20} + 763135551233075040t^{19} + 2372133882719751936t^{18} + 4025675628267611840t^{17} + 4184109756639291680t^{16} + 2906550748615536400t^{15} + 1493036856146122082t^{14} + 578683548147080154t^{13} + 113526827808735304t^{12} - 21520078670376065t^{11} - 13012676979302920t^{10} - 1124816653084680t^{9} - 388954174695472t^{8} - 184196517739311t^{7} - 9401129995162t^{6} + 5764377149264t^{5} + 256224691592t^{4} - 12523793512t^{3} - 981951888t^{2} - 45982404t - 566112 \right)D^{2} + 2 t ^{2} \left( 55936314129991680t^{20} + 412216856630852640t^{19} + 1270041315101141184t^{18} + 2129579512347169920t^{17} + 2186263454755156184t^{16} + 1505640555574779144t^{15} + 769970909646020596t^{14} + 290148852081314076t^{13} + 45931885929518396t^{12} - 18382238766224437t^{11} - 8763592223771542t^{10} - 872014164552508t^{9} - 146607577801184t^{8} - 71095363533792t^{7} - 3581545906520t^{6} + 2123970307132t^{5} + 169365653120t^{4} - 2481153680t^{3} - 275558528t^{2} - 13372320t - 118656 \right)D + 24 t ^{2} \left( 1902595718707200t^{20} + 13969399626189600t^{19} + 42773159655800640t^{18} + 71105797956688704t^{17} + 72347510243574248t^{16} + 49520003876387160t^{15} + 25265354571607144t^{14} + 9341873433284560t^{13} + 1233048786785792t^{12} - 746297507198261t^{11} - 317378761560338t^{10} - 30053746637082t^{9} - 3273811652280t^{8} - 1894257945142t^{7} - 64319610884t^{6} + 55225738900t^{5} + 5243762880t^{4} - 45130440t^{3} - 6083760t^{2} - 273960t - 1728 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 5)

|λ| = 10.09017 λ ≈ 10.09017; multiplicity 1; Jordan blocks 1 λ ≈ -6.905694 - 3.616011i; multiplicity 1; Jordan blocks 1 λ ≈ -6.905694 + 3.616011i; multiplicity 1; Jordan blocks 1 λ ≈ 2.721219; multiplicity 1; Jordan blocks 1 λ ≈ 0.5962467 - 1.333665i; multiplicity 1; Jordan blocks 1 λ ≈ 0.5962467 + 1.333665i; multiplicity 1; Jordan blocks 1 λ ≈ -1.102323; multiplicity 1; Jordan blocks 1 λ ≈ -1.09017; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$10.0901699$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.09017$11
$-6.905694 - 3.616011\mathrm{i}$11
$-6.905694 + 3.616011\mathrm{i}$11
$2.721219$11
$0.5962467 - 1.333665\mathrm{i}$11
$0.5962467 + 1.333665\mathrm{i}$11
$-1.102323$11
$-1.09017$11
$-1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} - 9 \lambda - 11\right) \left(\lambda^{6} + 11 \lambda^{5} + 23 \lambda^{4} - 156 \lambda^{3} + 60 \lambda^{2} - 81 \lambda - 389\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} - 9 \lambda - 11\right) \left(\lambda^{6} + 11 \lambda^{5} + 23 \lambda^{4} - 156 \lambda^{3} + 60 \lambda^{2} - 81 \lambda - 389\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 6 & 9 & 3 & 6 & 0 & 8 & 20 \\ 1 & -1 & 0 & 6 & 2 & 4 & 0 & 0 & 8 \\ 2 & 1 & 0 & 0 & 4 & 6 & 6 & 6 & 0 \\ 0 & 2 & 1 & -1 & 0 & 0 & 4 & 4 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 & 0 & 4 & 6 \\ 0 & 0 & 0 & 5 & 1 & \frac{3}{2} & -1 & 0 & 6 \\ 0 & 0 & 0 & 0 & 2 & 3 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0\end{pmatrix}\notag\end{equation}$$