Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-26-12-b

·

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

Defining data
Ambient
ℙ2P2 × ℙ3P3
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
E^(2w1 + 3w3)
Bundle
O_1(2) box O_2(3)
ZL62 label
2030.x319
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 12
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 26
$\chi(X, \mathrm{T}_X)$ = -96

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 26
$\mathrm{c}_1^2\mathrm{c}_2$ = 80
$\mathrm{c}_2^2$ = 249
$\mathrm{c}_1\mathrm{c}_3$ = -70
$\mathrm{c}_4$ = 251
Hodge diamond
1
00
0 2 0
0 0 0 0
0 30 185 30 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 96 0 96 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 150 0 0 138 0 12
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 4 0 4 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 7t + 10t^{2} + 7t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(735)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(726)

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( 8t + 1 \right) ^{5} \left( 784t^{2} + 42t + 1 \right) \left( 919t^{2} + 107t - 1 \right) \left( 13523161796214784t^{10} + 11182512458304512t^{9} + 4186862970958080t^{8} + 971133369399536t^{7} + 158778617836064t^{6} + 18787696277960t^{5} + 1520201366576t^{4} + 74429874963t^{3} + 1727532186t^{2} + 6932628t + 2236 \right)D^{6} + \left( 8t + 1 \right) ^{4} \left( 1636888508896295258161152t^{15} + 1682267830674888487862272t^{14} + 799089736014162981240832t^{13} + 237372292945742757246976t^{12} + 50056461848096386294528t^{11} + 7910052317291493311456t^{10} + 941000083784219640016t^{9} + 81993749964162148032t^{8} + 4995661582493387016t^{7} + 199431918826852758t^{6} + 4731172486660149t^{5} + 57768469636733t^{4} + 422474369267t^{3} + 6668489572t^{2} + 20820244t + 4472 \right)D^{5} + \left( 8t + 1 \right) ^{3} \left( 109125900593086350544076800t^{16} + 117568639908123381176729600t^{15} + 58862062752831213413859328t^{14} + 18520118468475562711760896t^{13} + 4155192054780174185293824t^{12} + 702734442677118031469312t^{11} + 90399079851659693374880t^{10} + 8665936974656827411792t^{9} + 596531311286696710000t^{8} + 28014782217369618840t^{7} + 832912338610714466t^{6} + 13851701566280081t^{5} + 101418909921916t^{4} + 89523665407t^{3} - 5574397780t^{2} - 14899836t - 2236 \right)D^{4} + t \left( 8t + 1 \right) ^{2} \left( 3666630259927701378280980480t^{16} + 4191540323033376847455846400t^{15} + 2236309699051401565579509760t^{14} + 751911492594340156676636672t^{13} + 180644147962803953493803008t^{12} + 32851681036458342343335936t^{11} + 4589244887217373113357568t^{10} + 486191295992114289458464t^{9} + 37976293860185463652176t^{8} + 2103061252111630607376t^{7} + 78194084034550191944t^{6} + 1796746149956357810t^{5} + 22285451268319333t^{4} + 122904444988724t^{3} + 331946802635t^{2} - 44684198t + 44032 \right)D^{3} + 2 \left( 8t + 1 \right) t ^{2} \left( 32406027440122922657569046528t^{16} + 39628011905366759593702588416t^{15} + 22692758077044594336379961344t^{14} + 8200937075365162103784603648t^{13} + 2118868502224464346557054976t^{12} + 415591734756261008604536832t^{11} + 63159684284728147076207616t^{10} + 7397709195200033469371904t^{9} + 654364064293826588852512t^{8} + 42428067994064425623952t^{7} + 1936507845713399634704t^{6} + 58692657280434839384t^{5} + 1079480837250252026t^{4} + 10488323577929045t^{3} + 48241634280818t^{2} + 142133049358t + 75497812 \right)D^{2} + 4 t ^{2} \left( 140798601981223732925989650432t^{17} + 185102702008196913071462023168t^{16} + 114270883520836973036396609536t^{15} + 44552955962831164481812299776t^{14} + 12416701842472271626846076928t^{13} + 2631800824597429733665734656t^{12} + 435307962553376486429515776t^{11} + 56274941698246061852391424t^{10} + 5611124878912561999380352t^{9} + 421994725311350529593440t^{8} + 23219609477248634050704t^{7} + 896067640669123462000t^{6} + 22788010601372139952t^{5} + 346955661645285198t^{4} + 2758788223570793t^{3} + 11069203191410t^{2} + 25959887020t + 22046960 \right)D + 32 t ^{2} \left( 7183602141899170047244369920t^{17} + 9282906124447500067751854080t^{16} + 5631913253617074813064445952t^{15} + 2159693761868620596809564160t^{14} + 592733322588392181380677632t^{13} + 123779718144012210372583424t^{12} + 20140236077157732383516672t^{11} + 2550959521900372812798848t^{10} + 247649637013484734773568t^{9} + 17984992266732152488352t^{8} + 945562995180158293872t^{7} + 34381711589177450088t^{6} + 807828194594041676t^{5} + 11051764629604022t^{4} + 76778907565673t^{3} + 276790584210t^{2} + 376582820t + 304096 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ3P3

|λ| = 114.9919 λ ≈ 114.9919; multiplicity 1; Jordan blocks 1 λ ≈ -21 - 18.52026i; multiplicity 1; Jordan blocks 1 λ ≈ -21 + 18.52026i; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 5; Jordan blocks 1, 4 λ ≈ -7.991869; multiplicity 1; Jordan blocks 1 5
Sector dimension
9
Spectral radius
$114.991869$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$114.9919$11
$-21 - 18.52026\mathrm{i}$11
$-21 + 18.52026\mathrm{i}$11
$-8$51, 4
$-7.991869$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 8\right)^{5} \left(\lambda^{2} + 42 \lambda + 784\right) \left(\lambda^{2} - 107 \lambda - 919\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 8\right)^{4} \left(\lambda^{2} + 42 \lambda + 784\right) \left(\lambda^{2} - 107 \lambda - 919\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 220 & 324 & 13068 & 5724 & 13152 & 456960 & 284400 & 3249760 \\ 1 & 0 & 22 & 612 & 324 & 600 & 25260 & 15966 & 178240 \\ 1 & 9 & 9 & 760 & 384 & 684 & 27072 & 16659 & 189600 \\ 0 & 1 & 1 & 9 & 3 & 28 & 732 & 504 & 6168 \\ 0 & 1 & 0 & 22 & -2 & 24 & 568 & 348 & 4656 \\ 0 & 0 & 1 & 9 & 9 & 0 & 580 & 384 & 3816 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{3} & 6 & 9 & \frac{440}{3} \\ 0 & 0 & 0 & 1 & 1 & -\frac{4}{9} & 24 & 3 & \frac{1064}{9} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$