Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-110-29-c

·

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

Defining data
Ambient
ℙ3P3 × Gr(2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w4 + w5) + E^(w1 + w7)
Bundle
U_2^v(1) + O_1(1) box Q_2
ZL62 label
3041.wFw5
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;5) \\ \det\mathcal{E} &\cong \mathcal{O}(3;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 29
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 110
$\chi(X, \mathrm{T}_X)$ = -19

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 110
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 121
$\mathrm{c}_1\mathrm{c}_3$ = 26
$\mathrm{c}_4$ = 23
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 17 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 + x1 + 19 0 x0 + 19 x1
$\mathrm{HH}^{ 3 }$ = x1 + 26 0 0 x1 + 15 11
$\mathrm{HH}^{ 4 }$ = 29 0 0 0 0 29
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 24t + 60t^{2} + 24t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(529)
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( 2t + 1 \right) \left( 3t + 1 \right) \left( 51409t^{7} + 92803t^{6} + 63626t^{5} + 20434t^{4} + 3016t^{3} + 166t^{2} - 1 \right) \left( 3002265366435332t^{13} + 10220043895309520t^{12} + 15732019567249988t^{11} + 14360700687383431t^{10} + 8579300652790842t^{9} + 3501011771885868t^{8} + 992589224997493t^{7} + 196416080120670t^{6} + 27246094678047t^{5} + 2693799604669t^{4} + 190576515503t^{3} + 8040936386t^{2} + 52140893t + 307890 \right)D^{6} + \left( 19447275988107321831288t^{22} + 110891451097737355333532t^{21} + 296345222627509258259256t^{20} + 492366680515202216219368t^{19} + 568915827201337833126480t^{18} + 484569124349229140424194t^{17} + 314710271444652558177782t^{16} + 159088759360763534842495t^{15} + 63377290894380580191554t^{14} + 20039652164282097449160t^{13} + 5048295097691365451172t^{12} + 1015261696741098950784t^{11} + 163295081947298354932t^{10} + 21056489067589225684t^{9} + 2179935686599918095t^{8} + 180170558886393584t^{7} + 11662246872613143t^{6} + 582307682145337t^{5} + 24224059257471t^{4} + 1006436741315t^{3} + 32198615479t^{2} + 153343779t + 615780 \right)D^{5} + \left( 162060633234227681927400t^{22} + 879953310001085749459540t^{21} + 2237957028355265475669856t^{20} + 3534648351678608933737772t^{19} + 3876015118463798666608607t^{18} + 3126316286816113409484076t^{17} + 1917828427770329947707163t^{16} + 913122360371791170364777t^{15} + 341644465403900478108070t^{14} + 101202927591658506382006t^{13} + 23843836434882600526265t^{12} + 4482282261125531788075t^{11} + 673754681025821899660t^{10} + 80866906007244683704t^{9} + 7643729615196604555t^{8} + 544448572282096312t^{7} + 26265289685105711t^{6} + 644130227968870t^{5} - 4033344252146t^{4} - 761388298773t^{3} - 28972921413t^{2} - 129035209t - 307890 \right)D^{4} + t \left( 680654659583756264095080t^{21} + 3550134919820365849274420t^{20} + 8669493149202697763172120t^{19} + 13134811045914481305697852t^{18} + 13796048736500351580096876t^{17} + 10638061729337396008033634t^{16} + 6224944729096590381494962t^{15} + 2820559348320300334297237t^{14} + 1002117558360102387015722t^{13} + 281445738941446314691232t^{12} + 62845367013197736297170t^{11} + 11212377391354559605240t^{10} + 1604510297689422700490t^{9} + 183886262060286294120t^{8} + 16584648844607973793t^{7} + 1120771550092255344t^{6} + 51526859583533161t^{5} + 1372317890149977t^{4} + 17709788580123t^{3} + 182075054361t^{2} + 1305084285t + 635373 \right)D^{3} + 2 t ^{2} \left( 751961338206816444143136t^{20} + 3798928843145858995503752t^{19} + 8982035806736789204179844t^{18} + 13163172756066203397066565t^{17} + 13354852831391142175146964t^{16} + 9929548894977778567839399t^{15} + 5591413536585300835655558t^{14} + 2433107779137021880327129t^{13} + 828726965930534606010688t^{12} + 222872839656385463630438t^{11} + 47651423506094241346825t^{10} + 8149611790749698059179t^{9} + 1119392734331071874766t^{8} + 122903914143112101553t^{7} + 10508822801033656885t^{6} + 657490010167361607t^{5} + 26926676078224687t^{4} + 612976318714863t^{3} + 7673338799509t^{2} + 103425480965t + 526084460 \right)D^{2} + 4 t ^{2} \left( 408392795750253758457048t^{20} + 2014207897882064953974932t^{19} + 4646960385109228018670916t^{18} + 6638998063433586778347331t^{17} + 6557792151718307626029978t^{16} + 4739407207434098871314205t^{15} + 2589494925879491951554278t^{14} + 1091379243695584888982355t^{13} + 359482580601409627922712t^{12} + 93404151036126415416262t^{11} + 19294147128502996299651t^{10} + 3190827069163186956444t^{9} + 423976936552746649704t^{8} + 44879456628079551678t^{7} + 3656590879465147635t^{6} + 212924028800997271t^{5} + 7819539300990927t^{4} + 153787144871721t^{3} + 1917010560216t^{2} + 25267870186t + 102630000 \right)D + 48 t ^{2} \left( 13890911420076658450920t^{20} + 67384398334082339973980t^{19} + 152828903709342748775988t^{18} + 214461911531319215807155t^{17} + 207832481444247153897506t^{16} + 147156595525249982422620t^{15} + 78649720423739208873256t^{14} + 32374660915312601138720t^{13} + 10400665317328340414862t^{12} + 2633415147878742471995t^{11} + 530015393864260554365t^{10} + 85442692123156852251t^{9} + 11064121702540170311t^{8} + 1136972898788765597t^{7} + 88942029203697429t^{6} + 4869352952465295t^{5} + 162742940745818t^{4} + 2838306339966t^{3} + 36197138826t^{2} + 396396246t + 1231560 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 5)

|λ| = 19.563 λ ≈ 19.563; multiplicity 1; Jordan blocks 1 λ ≈ -5.01516 - 7.88819i; multiplicity 1; Jordan blocks 1 λ ≈ -5.01516 + 7.88819i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.892736; multiplicity 1; Jordan blocks 1 λ ≈ -2.467817 - 0.1012637i; multiplicity 1; Jordan blocks 1 λ ≈ -2.467817 + 0.1012637i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.704313; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$19.563003$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.563$11
$-5.01516 - 7.88819\mathrm{i}$11
$-5.01516 + 7.88819\mathrm{i}$11
$-3$11
$-2.892736$11
$-2.467817 - 0.1012637\mathrm{i}$11
$-2.467817 + 0.1012637\mathrm{i}$11
$-2$11
$-1.704313$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{7} - 166 \lambda^{5} - 3016 \lambda^{4} - 20434 \lambda^{3} - 63626 \lambda^{2} - 92803 \lambda - 51409\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{7} - 166 \lambda^{5} - 3016 \lambda^{4} - 20434 \lambda^{3} - 63626 \lambda^{2} - 92803 \lambda - 51409\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 14 & 120 & 54 & 99 & 300 & 508 & 1025 \\ 1 & -1 & 5 & 4 & 0 & 10 & 45 & 63 & 200 \\ 1 & 1 & 0 & 24 & 14 & 14 & 78 & 129 & 220 \\ 0 & 1 & 2 & 0 & 0 & 5 & 10 & 14 & 60 \\ 0 & 1 & -1 & 4 & -1 & 2 & -6 & -4 & -15 \\ 0 & 0 & 1 & 1 & 1 & -2 & 10 & 14 & 30 \\ 0 & 0 & 0 & \frac{32}{11} & \frac{31}{11} & -\frac{14}{11} & \frac{29}{11} & \frac{111}{11} & 0 \\ 0 & 0 & 0 & \frac{7}{11} & -\frac{8}{11} & \frac{32}{11} & -\frac{5}{11} & -\frac{40}{11} & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{5} & \frac{12}{5} & 0\end{pmatrix}\notag\end{equation}$$