Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-164-39-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w4) + E^(2w3) + E^(w2 + w3)
Bundle
O_2(1) + Sym^2 U_2^v + Q_1 box U_2^v
ZL62 label
20601.3Aw7
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 164
$\chi(X, \mathrm{T}_X)$ = -7

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 164
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 116
$\mathrm{c}_1\mathrm{c}_3$ = 44
$\mathrm{c}_4$ = 20
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 12 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 + x2 + 7 0 x0 + 7 x2
$\mathrm{HH}^{ 3 }$ = x1 + 25 0 0 x1 25
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 37 0 0 x1 - x2 - 2 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 94t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(398)
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( 4t + 1 \right) \left( 8t + 1 \right) \left( 320t^{4} - 384t^{3} + 16t^{2} - 12t + 1 \right) \left( 35061760000t^{15} + 241467392000t^{14} - 337576755200t^{13} - 484746117120t^{12} - 1016875039232t^{11} + 1767452563200t^{10} + 502945840064t^{9} - 1013300773360t^{8} - 144072950672t^{7} + 95894873824t^{6} + 7381803904t^{5} - 5306741476t^{4} - 197648056t^{3} - 11888640t^{2} - 32182t - 1599 \right)D^{6} + 2 \left( 1974678323200000t^{22} + 12542436966400000t^{21} - 40273566498816000t^{20} - 4157574375014400t^{19} - 36786296442060800t^{18} + 241726918247251968t^{17} - 172433539419848704t^{16} - 125985529897771008t^{15} + 140813612374910976t^{14} + 10609702414924800t^{13} - 27190222800811264t^{12} - 1268316161081088t^{11} + 2323290187858400t^{10} + 71155757117040t^{9} - 85463978556360t^{8} - 12044352736072t^{7} - 698200322688t^{6} - 20881074076t^{5} - 18209350000t^{4} - 505681420t^{3} - 24253883t^{2} - 49872t - 1599 \right)D^{5} + \left( 17682346803200000t^{22} + 129929681305600000t^{21} - 346323734757376000t^{20} - 167039445709619200t^{19} - 571425002276782080t^{18} + 2587545787589787648t^{17} - 1269588904275615744t^{16} - 1690573231736049664t^{15} + 1274877302102991872t^{14} + 205580918145104640t^{13} - 225340686833681920t^{12} - 5041064966511488t^{11} + 22084631583736672t^{10} - 496358663616688t^{9} - 696234642735152t^{8} - 67434921167792t^{7} - 5033205221320t^{6} - 255194418524t^{5} + 40014123052t^{4} + 1011119904t^{3} + 46750850t^{2} + 108395t + 1599 \right)D^{4} + 2 t \left( 20644364288000000t^{21} + 167987222937600000t^{20} - 396727629643776000t^{19} - 318955715139993600t^{18} - 964536711656243200t^{17} + 3481390156949258240t^{16} - 1053641205421694976t^{15} - 2717860376067129344t^{14} + 1450359025550270976t^{13} + 424372110879061760t^{12} - 260976475195262848t^{11} - 15054658852310496t^{10} + 28467564923318848t^{9} - 1304030496656944t^{8} - 692993892964408t^{7} - 54458540850832t^{6} - 2364548361864t^{5} - 101214409536t^{4} - 4002506620t^{3} - 59309920t^{2} - 1807837t - 429 \right)D^{3} + 4 t ^{2} \left( 13261760102400000t^{20} + 116376305664000000t^{19} - 253952361103360000t^{18} - 267887290879180800t^{17} - 802663908476190720t^{16} + 2526184074581825536t^{15} - 355610389071086592t^{14} - 2234776672686845696t^{13} + 842027264313958080t^{12} + 404302357851952448t^{11} - 176400483824687360t^{10} - 23622228053522976t^{9} + 21570562360640680t^{8} - 989892792549344t^{7} - 317843749540100t^{6} - 24760877890760t^{5} - 1594170110606t^{4} - 35711153850t^{3} - 1404610804t^{2} - 35316748t - 706963 \right)D^{2} + 32 t ^{2} \left( 1110756556800000t^{20} + 10328261427200000t^{19} - 21394345279488000t^{18} - 26456522340352000t^{17} - 81034890925875200t^{16} + 235318119875260672t^{15} - 1157625579764352t^{14} - 226957279300299296t^{13} + 57939863214637256t^{12} + 44935093728083216t^{11} - 15933780278555032t^{10} - 3585459593405496t^{9} + 2188946698267944t^{8} - 76423679265658t^{7} - 16480121399443t^{6} - 1599410968599t^{5} - 160280654365t^{4} - 1828433042t^{3} - 84878032t^{2} - 2366626t - 36777 \right)D + 32 t ^{2} \left( 302933606400000t^{20} + 2947230924800000t^{19} - 5896335687680000t^{18} - 8022708749107200t^{17} - 25350490122895360t^{16} + 70335212239648256t^{15} + 7444759815700736t^{14} - 72031802034723776t^{13} + 11451055714253616t^{12} + 15109642267488576t^{11} - 4702787047325016t^{10} - 1477503824582336t^{9} + 716451973276132t^{8} - 15793147934108t^{7} - 2036083980016t^{6} - 364095593988t^{5} - 50950858992t^{4} - 397704252t^{3} - 22046349t^{2} - 511797t - 6396 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 7)

|λ| = 12.91482 λ ≈ 12.91482; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 1; Jordan blocks 1 λ ≈ -0.8801913 - 5.341209i; multiplicity 1; Jordan blocks 1 λ ≈ -0.8801913 + 5.341209i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ 1; multiplicity 1; Jordan blocks 1 λ ≈ 0.8455628; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
10
Spectral radius
$12.9148198$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.91482$11
$-8$11
$-0.8801913 - 5.341209\mathrm{i}$11
$-0.8801913 + 5.341209\mathrm{i}$11
$-4$11
$1$11
$0.8455628$11
$0$31, 2

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 4\right) \left(\lambda + 8\right) \left(\lambda - 1\right) \left(\lambda^{4} - 12 \lambda^{3} + 16 \lambda^{2} - 384 \lambda + 320\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right) \left(\lambda + 8\right) \left(\lambda - 1\right) \left(\lambda^{4} - 12 \lambda^{3} + 16 \lambda^{2} - 384 \lambda + 320\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 12 & 24 & 12 & 12 & 24 & 96 & 128 & 160 \\ 1 & 0 & 2 & 4 & 0 & 8 & 12 & -12 & -12 & 64 \\ 1 & 0 & 0 & 8 & 4 & 8 & 12 & 24 & 36 & 64 \\ 0 & 1 & 1 & 0 & -1 & 2 & 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 & 1 & 2 & 2 & 12 & 16 & 0 \\ 0 & 0 & 1 & 0 & 1 & 2 & 2 & 4 & 8 & 24 \\ 0 & 0 & 1 & 0 & 0 & -2 & -2 & 4 & 4 & 0 \\ 0 & 0 & 0 & -1 & -2 & 4 & 3 & -2 & -2 & 0 \\ 0 & 0 & 0 & 3 & 2 & -1 & 1 & 2 & 2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$