Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-100-27-a

·

This is a Fano fourfold of K3 type, appearing as case K3-38 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × Gr(2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
3 copies of E^(w4) + E^(w1 + w2 + w4)
Bundle
3 copies of O_3(1) + O_1(1) box O_2(1) box O_3(1)
ZL62 label
1141.333x4100
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 100
$\chi(X, \mathrm{T}_X)$ = -18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 100
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 32
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 3 0
0 0 0 0
0 1 22 1 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 + 18 0 x0 + 18 x1
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 + 13 2
$\mathrm{HH}^{ 4 }$ = 28 0 0 1 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 54t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(637)
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( 5t + 1 \right) ^{3} \left( 31t^{2} + t - 1 \right) \left( 199t^{4} + 218t^{3} + 117t^{2} + 18t - 1 \right) \left( 386715506181t^{10} + 592098252233t^{9} + 385956844920t^{8} + 138830740165t^{7} + 29791929656t^{6} + 3835321985t^{5} + 279393071t^{4} + 9157568t^{3} - 111900t^{2} - 19404t + 60 \right)D^{6} + \left( 5t + 1 \right) ^{2} \left( 250493035551211845t^{17} + 659102009476961983t^{16} + 805197022502643317t^{15} + 592972864486735778t^{14} + 287862037981043565t^{13} + 95167380919996415t^{12} + 21407965325990891t^{11} + 3148752074164442t^{10} + 262173676113971t^{9} + 2940586540825t^{8} - 2125846356065t^{7} - 301196594732t^{6} - 23719302084t^{5} - 1062915486t^{4} - 17849740t^{3} + 124656t^{2} + 58032t - 120 \right)D^{5} + \left( 5t + 1 \right) \left( 10437209814633826875t^{18} + 27649520472514422800t^{17} + 33873338046699040762t^{16} + 25201996323236808012t^{15} + 12594564622463821853t^{14} + 4423223081532803244t^{13} + 1111379713070725053t^{12} + 199604738920475364t^{11} + 25134058959776437t^{10} + 2119794715870836t^{9} + 109086759418325t^{8} + 3159232449656t^{7} + 209025194082t^{6} + 36841956664t^{5} + 3151865831t^{4} + 105169964t^{3} + 561828t^{2} - 34872t + 60 \right)D^{4} + t \left( 219181406107310364375t^{18} + 593117875037049935875t^{17} + 742117385833997433100t^{16} + 566920020301766363081t^{15} + 294123713257936220174t^{14} + 109090179628195440285t^{13} + 29661978734806411818t^{12} + 5965853366041024885t^{11} + 883891715438543090t^{10} + 94563122490293537t^{9} + 6939023128343158t^{8} + 297546895682541t^{7} + 1529255977391t^{6} - 574238755582t^{5} - 19884184726t^{4} + 503212896t^{3} + 12083940t^{2} - 717840t - 216 \right)D^{3} + 2 t ^{2} \left( 242143267699504783500t^{17} + 628748773093166284850t^{16} + 751571783517816358795t^{15} + 546784579224265818241t^{14} + 269638278506160830594t^{13} + 94983441988263877385t^{12} + 24535563049030034217t^{11} + 4694507881148090992t^{10} + 663251284569286142t^{9} + 67868537416193200t^{8} + 4768958514550484t^{7} + 192746652493548t^{6} + 102197904203t^{5} - 454669725217t^{4} - 20055095249t^{3} - 180674544t^{2} - 3214566t - 216780 \right)D^{2} + 4 t ^{2} \left( 131508843664386218625t^{17} + 330926594027854380775t^{16} + 382063836066411319625t^{15} + 267680449278424724882t^{14} + 126806020124431430439t^{13} + 42824058544629919441t^{12} + 10588171155681229654t^{11} + 1936445698991246429t^{10} + 261052157520282935t^{9} + 25382431337641662t^{8} + 1670208904160113t^{7} + 58660773746619t^{6} - 780818320395t^{5} - 197886217960t^{4} - 8550886268t^{3} - 146787660t^{2} - 2547300t - 35520 \right)D + 24 t ^{2} \left( 8946179841114708750t^{17} + 22027190439321958250t^{16} + 24820171199285595010t^{15} + 16929137771370012766t^{14} + 7788123962564890387t^{13} + 2547984593361751603t^{12} + 608792737765595881t^{11} + 107294860156930976t^{10} + 13882890053794666t^{9} + 1284823765714248t^{8} + 78447506170817t^{7} + 2207605395551t^{6} - 90575441727t^{5} - 12032794024t^{4} - 492133244t^{3} - 9785628t^{2} - 157188t - 960 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × Gr(2, 5)

|λ| = 23.41093 λ ≈ 23.41093; multiplicity 1; Jordan blocks 1 λ ≈ 6.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -5.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -5; multiplicity 4; Jordan blocks 1, 1, 2 λ ≈ -3.230593; multiplicity 1; Jordan blocks 1 λ ≈ -1.09017 - 1.201132i; multiplicity 1; Jordan blocks 1 λ ≈ -1.09017 + 1.201132i; multiplicity 1; Jordan blocks 1 2 2 4
Sector dimension
12
Spectral radius
$23.4109327$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$23.41093$11
$6.09017$21, 1
$-5.09017$21, 1
$-5$41, 1, 2
$-3.230593$11
$-1.09017 - 1.201132\mathrm{i}$11
$-1.09017 + 1.201132\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{4} \left(\lambda^{2} - \lambda - 31\right)^{2} \left(\lambda^{4} - 18 \lambda^{3} - 117 \lambda^{2} - 218 \lambda - 199\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right)^{2} \left(\lambda^{2} - \lambda - 31\right) \left(\lambda^{4} - 18 \lambda^{3} - 117 \lambda^{2} - 218 \lambda - 199\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 32 & 18 & 78 & 78 & 90 & 240 & 248 & 248 & 680 \\ 1 & -1 & 1 & 3 & 6 & 26 & 6 & 12 & 60 & 66 & 36 & 152 \\ 1 & 1 & -1 & 3 & 6 & 6 & 26 & 12 & 60 & 36 & 66 & 152 \\ 1 & 1 & 1 & 2 & 2 & 12 & 12 & 16 & 30 & 36 & 36 & 96 \\ 0 & 1 & 1 & 0 & -2 & 3 & 3 & 0 & 20 & 12 & 12 & 42 \\ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 2 & 10 & 12 & 4 & 24 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 10 & 4 & 12 & 24 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & \frac{5}{2} & \frac{5}{2} & 0 & 5 & 8 & 8 & 18 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -\frac{4}{5} & 0 & \frac{6}{5} & \frac{6}{5} & \frac{32}{5} \\ 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 0 & 2 & \frac{5}{2} & 0 & 2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & \frac{5}{2} & 2 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$