Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-198-46-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ2P2
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_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + w3 + w6)
Bundle
O_1(1) box O_2(1) box Q_3
ZL62 label
202020.x4B00
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 46
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 198
$\chi(X, \mathrm{T}_X)$ = -2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 198
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 99
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 9
Hodge diamond
1
00
0 3 0
0 1 1 0
0 0 5 0 0
0 1 1 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 + 40 0 0 x1 + 6 34
$\mathrm{HH}^{ 4 }$ = 46 0 0 0 0 46
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 41t + 114t^{2} + 41t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(322)
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( 6t + 1 \right) \left( 12t - 1 \right) \left( 9t^{2} + 3t + 1 \right) \left( 36t^{2} - 6t + 1 \right) \left( 8t^{2} + 4t + 1 \right) ^{2} \left( 1254794084352t^{14} + 2438963675136t^{13} + 1702474454016t^{12} + 554016858624t^{11} + 253357103232t^{10} + 243982611648t^{9} + 176837691456t^{8} + 69470993520t^{7} + 15650269680t^{6} + 2031988356t^{5} + 159489018t^{4} + 12071092t^{3} + 454975t^{2} + 8845t - 308 \right)D^{6} + \left( 8t^{2} + 4t + 1 \right) \left( 4917668515160260608t^{22} + 12648186727967490048t^{21} + 13520959978141384704t^{20} + 8215188969844703232t^{19} + 3975454201870614528t^{18} + 2432613948819111936t^{17} + 1844746741036929024t^{16} + 1133785122493068288t^{15} + 500754969677822976t^{14} + 158732536967843328t^{13} + 36670735147503744t^{12} + 6283333054425600t^{11} + 812924491137984t^{10} + 81025002377136t^{9} + 6724853998656t^{8} + 730574313792t^{7} + 118670016942t^{6} + 14686151486t^{5} + 958091105t^{4} + 66736145t^{3} + 2065555t^{2} + 37536t - 924 \right)D^{5} + \left( 327844567677350707200t^{24} + 975208494850753167360t^{23} + 1249639225770003922944t^{22} + 939034956416576126976t^{21} + 527459203356777381888t^{20} + 311146971620242194432t^{19} + 226480937045286813696t^{18} + 154546053203947474944t^{17} + 82025683325427234816t^{16} + 32629771244814041088t^{15} + 9779870679336552960t^{14} + 2229720805729354752t^{13} + 389577489334207488t^{12} + 52539704030314368t^{11} + 5494388797486848t^{10} + 434248128623856t^{9} + 19821376685280t^{8} - 899350314404t^{7} - 307851651402t^{6} - 32504582754t^{5} - 1899547249t^{4} - 99410484t^{3} - 2155364t^{2} - 31663t + 616 \right)D^{4} + t \left( 1376947184244872970240t^{23} + 3990519094560172277760t^{22} + 4886438987302761922560t^{21} + 3459250522974498324480t^{20} + 1894459337707123113984t^{19} + 1205771891758354464768t^{18} + 949556539302252576768t^{17} + 648409037112809570304t^{16} + 330037650870958485504t^{15} + 123657507514257564672t^{14} + 34585022669456655360t^{13} + 7322964758606237184t^{12} + 1188711691604083584t^{11} + 150407797025446272t^{10} + 15162630535387392t^{9} + 1236790191422064t^{8} + 77642335113744t^{7} + 3375297956024t^{6} + 69705623410t^{5} - 3322482890t^{4} - 290026081t^{3} - 9391617t^{2} - 275783t + 1036 \right)D^{3} + 2 t \left( 1521198794022907281408t^{23} + 4319522077672162197504t^{22} + 5101818070103416111104t^{21} + 3448203168167470301184t^{20} + 1872870035702912188416t^{19} + 1283248859943894958080t^{18} + 1062448016339768131584t^{17} + 717898885863925082112t^{16} + 351672994439158809600t^{15} + 125266087572917497344t^{14} + 33082727005532740608t^{13} + 6596658436349938176t^{12} + 1011976763882974848t^{11} + 122677204111176672t^{10} + 12094582991521968t^{9} + 980752987196568t^{8} + 60770243145216t^{7} + 2438899039404t^{6} + 25903300408t^{5} - 3307017066t^{4} - 233406483t^{3} - 4165665t^{2} - 63085t + 154 \right)D^{2} + 8 t ^{2} \left( 413084155273461891072t^{22} + 1155252541624248434688t^{21} + 1327705633626624884736t^{20} + 867021402027978915840t^{19} + 471500628923306704896t^{18} + 343184500934501695488t^{17} + 292769723936812142592t^{16} + 195362736865973332992t^{15} + 92806254530935045632t^{14} + 31780894977091948032t^{13} + 8027604923420797056t^{12} + 1528588956206658816t^{11} + 225052290521889696t^{10} + 26565489677685744t^{9} + 2586224450796912t^{8} + 206919728234400t^{7} + 12366363450288t^{6} + 452681073474t^{5} + 1143774391t^{4} - 723514745t^{3} - 46106835t^{2} - 595785t - 2992 \right)D + 288 t ^{3} \left( 4683493823962152960t^{21} + 12962360573181296640t^{20} + 14617472738451259392t^{19} + 9322150563995516928t^{18} + 5092235449471107072t^{17} + 3869736467347574784t^{16} + 3358431555055073280t^{15} + 2218332472365947904t^{14} + 1031126263908991488t^{13} + 343433193828868224t^{12} + 84044257682306112t^{11} + 15487994251621440t^{10} + 2217089075419248t^{9} + 257629951345056t^{8} + 24892569966900t^{7} + 1960787536454t^{6} + 112548665510t^{5} + 3799389783t^{4} - 10378536t^{3} - 6440360t^{2} - 376356t - 4158 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ2P2

|λ| = 12 λ ≈ 12; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 2; Jordan blocks 1, 1 λ ≈ 3 - 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ 3 + 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ -1.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2 + 2i; multiplicity 2; Jordan blocks 1, 1 2 2 2 2 2
Sector dimension
13
Spectral radius
$12$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12$11
$-6$21, 1
$3 - 5.196152\mathrm{i}$11
$3 + 5.196152\mathrm{i}$11
$-1.5 - 2.598076\mathrm{i}$21, 1
$-1.5 + 2.598076\mathrm{i}$21, 1
$-2 - 2\mathrm{i}$21, 1
$-2 + 2\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda - 12\right) \left(\lambda^{2} + 3 \lambda + 9\right)^{2} \left(\lambda^{2} + 4 \lambda + 8\right)^{2} \left(\lambda^{2} - 6 \lambda + 36\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda - 12\right) \left(\lambda^{2} + 3 \lambda + 9\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{2} - 6 \lambda + 36\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 0 & 12 & 3 & 9 & 3 & 6 & 24 & 12 & 24 & 60 \\ 1 & -1 & 2 & 0 & 4 & 0 & 2 & 0 & 2 & 3 & 3 & 3 & 16 \\ 1 & 2 & -1 & 0 & 4 & 0 & 2 & 0 & 2 & 3 & 0 & 6 & 16 \\ 2 & 1 & 1 & 0 & 8 & 6 & 4 & 6 & -2 & 12 & 6 & 0 & 8 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 2 & -1 & 1 & -1 & -1 & -1 & 2 & 0 & 4 & 2 & -2 & -6 \\ 0 & 1 & 1 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 12 \\ 0 & 0 & 3 & 1 & 3 & 2 & 2 & -1 & 0 & 4 & 2 & 0 & 6 \\ 0 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 0 & 8 & 4 & 6 & 12 \\ 0 & 0 & 0 & 0 & 3 & 2 & 3 & -1 & -1 & -1 & 1 & -2 & -4 \\ 0 & 0 & 0 & 0 & 0 & -1 & -3 & 5 & 2 & 2 & -2 & 4 & 8 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 2 & 1 & 1 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{3}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$