Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-99-28-a

·

This is a Fano fourfold of K3 type, appearing as case C-2 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ2P2 × ℙ5P5
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(3w3) + E^(w2 + w3)
Bundle
O_2(3) + Q_1 box O_2(1)
ZL62 label
2050.Gw6
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 28
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 99
$\chi(X, \mathrm{T}_X)$ = -29

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 99
$\mathrm{c}_1^2\mathrm{c}_2$ = 126
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 18
$\mathrm{c}_4$ = 27
Hodge diamond
1
00
0 2 0
0 1 1 0
0 1 23 1 0
0 1 1 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 + x2 + 29 0 x0 + 29 x2
$\mathrm{HH}^{ 3 }$ = x1 + 17 0 0 x1 17
$\mathrm{HH}^{ 4 }$ = x1 - x2 + x3 + 4 0 0 x1 - x2 - 24 x3 28
$\mathrm{HH}^{ 5 }$ = x3 - 10 0 0 x3 - 10 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 23t + 51t^{2} + 23t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(691)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(688)

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( 30t - 1 \right) \left( 6t + 1 \right) ^{2} \left( 333t^{2} - 33t + 1 \right) \left( 40t^{2} + 12t + 1 \right) ^{2} \left( 382276308781440t^{11} + 335267452019520t^{10} + 172667987165088t^{9} + 53951948881056t^{8} + 9107896437144t^{7} + 791872667316t^{6} + 32142418158t^{5} + 514328896t^{4} + 6563039t^{3} + 1042573t^{2} - 25500t + 44 \right)D^{6} + \left( 6t + 1 \right) \left( 40t^{2} + 12t + 1 \right) \left( 19247459236621991424000t^{17} + 21456293810753440051200t^{16} + 13062743008187706662400t^{15} + 5058649621625545198080t^{14} + 1198020567729587238144t^{13} + 164959838250204559104t^{12} + 11824512162371278272t^{11} + 243525574310201280t^{10} - 20920016536988112t^{9} - 1598265748679760t^{8} - 41996433961242t^{7} + 656847689074t^{6} + 58280994897t^{5} + 6823861313t^{4} - 73399157t^{3} + 4606912t^{2} - 75180t + 88 \right)D^{5} + \left( 38494918473243982848000000t^{20} + 57904965964451820257280000t^{19} + 45244956528871147313049600t^{18} + 23091386960901905888993280t^{17} + 8038200419693703310740480t^{16} + 1923952806011935500254208t^{15} + 316509009941835874384128t^{14} + 35373981124940929665408t^{13} + 2595086552680755416640t^{12} + 114227440653437395488t^{11} + 2155654621204710336t^{10} - 42313664642423496t^{9} - 4669140229495664t^{8} - 369034030358262t^{7} - 22761498310596t^{6} - 497403512749t^{5} - 100951872t^{4} - 104906922t^{3} - 3265263t^{2} + 44864t - 44 \right)D^{4} + t \left( 161678657587624727961600000t^{19} + 233393531133620214005760000t^{18} + 178087629462130787650560000t^{17} + 88879723733342972098252800t^{16} + 29944982673641925051264000t^{15} + 6863932636031067560547840t^{14} + 1075625145357476525314560t^{13} + 114926422258858208479488t^{12} + 8238768820332861445632t^{11} + 383037267832249082688t^{10} + 11130757678239567552t^{9} + 233844424682898000t^{8} + 5836884315990932t^{7} - 50733161263514t^{6} - 13103002234084t^{5} + 47604845345t^{4} + 8187389965t^{3} - 129780419t^{2} + 1263940t + 284 \right)D^{3} + 2 t ^{2} \left( 178616421715852080414720000t^{18} + 249554756536755455397888000t^{17} + 187009292471170870247270400t^{16} + 91618823935949044782551040t^{15} + 29987933482145995794247680t^{14} + 6612479097740903359231488t^{13} + 991373173564481959629312t^{12} + 101409631044904131398784t^{11} + 7040347056195794569152t^{10} + 328958331051295340640t^{9} + 10678993587792737784t^{8} + 295322234212290552t^{7} + 7415087604433812t^{6} - 55363495373876t^{5} - 8819496946216t^{4} + 88272106823t^{3} + 3819383331t^{2} - 30373537t + 374924 \right)D^{2} + 8 t ^{2} \left( 48503597276287418388480000t^{18} + 66117418350339358660608000t^{17} + 48894803802792477642931200t^{16} + 23608801162795434267571200t^{15} + 7546064615483884248648960t^{14} + 1611359549062209848335104t^{13} + 232802451391851618217920t^{12} + 22940609519586004629120t^{11} + 1545417295989471127152t^{10} + 71739398395984311072t^{9} + 2451034412825903064t^{8} + 74838376238641740t^{7} + 1788296434757076t^{6} - 18095311612967t^{5} - 1626733247861t^{4} + 21380798529t^{3} + 465058142t^{2} - 779502t + 29480 \right)D + 864 t ^{2} \left( 183309135586876108800000t^{18} + 245664964082108897280000t^{17} + 180044177799437641497600t^{16} + 86038111833894763937280t^{15} + 27023133094133350769280t^{14} + 5633811116128377353088t^{13} + 791470369596998531808t^{12} + 75774542672460005280t^{11} + 4981706249385968712t^{10} + 229184850242750712t^{9} + 8039153456392550t^{8} + 256979748432212t^{7} + 5873575232517t^{6} - 66006892378t^{5} - 4161948590t^{4} + 71458312t^{3} + 866939t^{2} + 4146t + 44 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ5P5

|λ| = 30 λ ≈ 30; multiplicity 1; Jordan blocks 1 λ ≈ 16.5 - 7.794229i; multiplicity 1; Jordan blocks 1 λ ≈ 16.5 + 7.794229i; multiplicity 1; Jordan blocks 1 λ ≈ -6 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6 + 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6; multiplicity 2; Jordan blocks 2 2 2 2
Sector dimension
9
Spectral radius
$30$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$30$11
$16.5 - 7.794229\mathrm{i}$11
$16.5 + 7.794229\mathrm{i}$11
$-6 - 2\mathrm{i}$21, 1
$-6 + 2\mathrm{i}$21, 1
$-6$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda - 30\right) \left(\lambda^{2} + 12 \lambda + 40\right)^{2} \left(\lambda^{2} - 33 \lambda + 333\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda - 30\right) \left(\lambda^{2} + 12 \lambda + 40\right) \left(\lambda^{2} - 33 \lambda + 333\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 108 & 18 & 18 & 774 & 504 & 432 & 3780 \\ 2 & 0 & 6 & 108 & -2 & 144 & 756 & 0 & 72 \\ 1 & 0 & 9 & 0 & 2 & 108 & 45 & 45 & 432 \\ 0 & 1 & 2 & 9 & 0 & 15 & 108 & -2 & 0 \\ 0 & 2 & 0 & 6 & 0 & 6 & 144 & 108 & 756 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 18 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 15 & 9 & 108 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0\end{pmatrix}\notag\end{equation}$$