Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-110-29-a

·

This is a Fano fourfold of K3 type, appearing as case K3-32 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Fl(1, 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_4ω1ω2ω3ω4
Bundle
2 copies of E^(w2) + E^(w1 + w2)
Bundle
2 copies of O(0,1) + O(1,1)
ZL62 label
42.11S
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 29
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 110
$\chi(X, \mathrm{T}_X)$ = -21

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 110
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 30
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 22 1 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 + 21 0 x0 + 21 x1
$\mathrm{HH}^{ 3 }$ = x1 + 22 0 0 x1 + 11 11
$\mathrm{HH}^{ 4 }$ = 30 0 0 1 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(559)
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) ^{3} \left( 93t^{3} + 44t^{2} + 17t - 1 \right) \left( 152t^{3} + 116t^{2} + 18t + 1 \right) \left( 271937084160t^{10} + 1138471295360t^{9} + 1478552039680t^{8} + 959032153984t^{7} + 363582611824t^{6} + 81201583288t^{5} + 9559355320t^{4} + 700761316t^{3} + 127494796t^{2} + 55306t + 6215 \right)D^{6} + 2 \left( 2t + 1 \right) ^{2} \left( 80726155055400960t^{17} + 457910413023974400t^{16} + 995558467194639360t^{15} + 1213075973210781696t^{14} + 959956278199829760t^{13} + 528761882876776704t^{12} + 209190741191377792t^{11} + 59988570500385568t^{10} + 12434967821695912t^{9} + 1879262033440860t^{8} + 225210989256660t^{7} + 25233890220702t^{6} + 2281154980716t^{5} + 86326621194t^{4} + 1252102785t^{3} + 258924668t^{2} + 33239t + 6215 \right)D^{5} + \left( 2t + 1 \right) \left( 2690871835180032000t^{18} + 16070062445122560000t^{17} + 37101084823448238080t^{16} + 48086188541349656576t^{15} + 40515797558089865216t^{14} + 23841514800345319424t^{13} + 10149029794651859712t^{12} + 3168537772067058944t^{11} + 727698017117399904t^{10} + 124243743675208736t^{9} + 16672241486472024t^{8} + 1963405354839736t^{7} + 196574301782908t^{6} + 11250620037508t^{5} + 113380027708t^{4} - 1060502976t^{3} - 499516613t^{2} - 861723t - 6215 \right)D^{4} + t \left( 22603323415512268800t^{18} + 142732430463226060800t^{17} + 353408400676698009600t^{16} + 494055009965339441152t^{15} + 450887583019702071296t^{14} + 288897936590871689216t^{13} + 134944143564294038016t^{12} + 46741650496122394880t^{11} + 12082053788739253184t^{10} + 2353771553977895712t^{9} + 359616491609558832t^{8} + 46941198299951352t^{7} + 5414603626895816t^{6} + 460134946279084t^{5} + 19319666711608t^{4} + 178802123784t^{3} + 2584783300t^{2} + 39688891t + 11526 \right)D^{3} + t ^{2} \left( 49942581260941393920t^{17} + 309356402056063057920t^{16} + 738066407467531755520t^{15} + 987473729125202186240t^{14} + 859458687966392514560t^{13} + 523629826910201178112t^{12} + 231614854355597224960t^{11} + 75532378903661871616t^{10} + 18294384763167119104t^{9} + 3359003582445139200t^{8} + 498714758832924576t^{7} + 65494165785699824t^{6} + 7308952181913640t^{5} + 538271800238992t^{4} + 16883127183308t^{3} + 118786560842t^{2} + 3710604077t + 27849470 \right)D^{2} + 16 t ^{2} \left( 3390498512326840320t^{17} + 20702356317071493120t^{16} + 48000985544454604800t^{15} + 62093333202744233984t^{14} + 52124076385835155456t^{13} + 30559931930485208576t^{12} + 12960091010471518080t^{11} + 4030449671590636096t^{10} + 927533755717763568t^{9} + 163213067399558616t^{8} + 23939731256195712t^{7} + 3152916927144300t^{6} + 333733245817300t^{5} + 21212931865340t^{4} + 502959526966t^{3} + 3977391597t^{2} + 130023458t + 615285 \right)D + 16 t ^{2} \left( 1383876943806873600t^{17} + 8367433102896537600t^{16} + 19015923627762647040t^{15} + 24020668511261306880t^{14} + 19654390405325795840t^{13} + 11211463754576509440t^{12} + 4611176899859666816t^{11} + 1384157386834819584t^{10} + 306562329730477856t^{9} + 52381463000478048t^{8} + 7654170785192280t^{7} + 1005703451802336t^{6} + 100563401829696t^{5} + 5586973296610t^{4} + 103027496385t^{3} + 1148662546t^{2} + 28029537t + 87010 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5)

|λ| = 19.50086 λ ≈ 19.50086; multiplicity 1; Jordan blocks 1 λ ≈ -8.134884 - 4.655448i; multiplicity 1; Jordan blocks 1 λ ≈ -8.134884 + 4.655448i; multiplicity 1; Jordan blocks 1 λ ≈ -1.250432 - 1.790374i; multiplicity 1; Jordan blocks 1 λ ≈ -1.250432 + 1.790374i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 λ ≈ -1.730231; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$19.5008645$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.50086$11
$-8.134884 - 4.655448\mathrm{i}$11
$-8.134884 + 4.655448\mathrm{i}$11
$-1.250432 - 1.790374\mathrm{i}$11
$-1.250432 + 1.790374\mathrm{i}$11
$-2$31, 2
$-1.730231$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{3} + 18 \lambda^{2} + 116 \lambda + 152\right) \left(\lambda^{3} - 17 \lambda^{2} - 44 \lambda - 93\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{3} + 18 \lambda^{2} + 116 \lambda + 152\right) \left(\lambda^{3} - 17 \lambda^{2} - 44 \lambda - 93\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 14 & 14 & 48 & 60 & 78 & 256 & 208 & 600 \\ 1 & -1 & 4 & 2 & 10 & 4 & 63 & 54 & 96 \\ 1 & 2 & -1 & 10 & 18 & 16 & 51 & 48 & 120 \\ 0 & 1 & 2 & -1 & 1 & 0 & 8 & 4 & 18 \\ 0 & 2 & 1 & 3 & -1 & 5 & 8 & 4 & 36 \\ 0 & 0 & 1 & 0 & 2 & -1 & 14 & 12 & 18 \\ 0 & 0 & 0 & 1 & \frac{19}{5} & \frac{7}{5} & \frac{11}{5} & \frac{28}{5} & \frac{42}{5} \\ 0 & 0 & 0 & 1 & -\frac{8}{5} & \frac{17}{10} & -\frac{2}{5} & -\frac{21}{5} & \frac{21}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & 2 & 0\end{pmatrix}\notag\end{equation}$$