Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-147-36-b

·

A degree-$6$ del Pezzo fibration over $\mathbb P^2$.

  • $\pi_1\colon X\to Y$ maps birationally onto a fourfold of degree $39$ in $\mathrm{Gr}(2,8)$; it contracts two disjoint planes, each with normal bundle $\mathcal O(-1)^{\oplus 2}$, to points and is an isomorphism elsewhere.

  • $\pi_2\colon X\to\mathbb P^2$ is a sextic del Pezzo fibration, of which the two planes above are sections; its double cover is $\mathbb P^1\times\mathbb P^1\to\mathbb P^2$, branched along a smooth conic, and its triple cover is a $\mathbb P^1$-bundle over a smooth plane cubic curve.

Defining data
Ambient
ℙ2P2 × Gr(2, 8)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w5) + E^(w2 + w3)
Bundle
Q_2^v(1) + Q_1 box U_2^v
ZL62 label
20701.4w8
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;8) \\ \det\mathcal{E} &\cong \mathcal{O}(2;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 147
$\chi(X, \mathrm{T}_X)$ = -6

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 147
$\mathrm{c}_1^2\mathrm{c}_2$ = 126
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 46
$\mathrm{c}_4$ = 7
Hodge diamond
1
00
0 2 0
0 1 1 0
0 0 5 0 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 + x1 + 6 0 x0 + 6 x1
$\mathrm{HH}^{ 3 }$ = x1 + 36 0 0 x1 + 18 18
$\mathrm{HH}^{ 4 }$ = 36 0 0 0 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 83t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
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( 8t^{2} + 4t + 1 \right) ^{2} \left( 11657t^{5} + 3474t^{4} + 480t^{3} - 9t^{2} + 12t - 1 \right) \left( 3932104531289760t^{15} + 9003535901031120t^{14} + 10522761791978760t^{13} + 6042326973716256t^{12} + 1909907476082196t^{11} + 222374923056596t^{10} + 11284119812044t^{9} + 68423074943284t^{8} + 47484802161904t^{7} + 14183979512228t^{6} + 2173075321502t^{5} + 175014470632t^{4} + 7696389827t^{3} + 257166869t^{2} + 468165t + 10143 \right)D^{6} + \left( 8t^{2} + 4t + 1 \right) \left( 15401078287138230059520t^{23} + 52026902951219821528320t^{22} + 87237568994540443684800t^{21} + 86795792719889951928288t^{20} + 55795756155434477767632t^{19} + 23884691725750034023592t^{18} + 6833580657272584133224t^{17} + 1648749956955320950524t^{16} + 789669558891247433352t^{15} + 521085996871292830100t^{14} + 246006559188616682828t^{13} + 78552968824005567888t^{12} + 17500899695618053044t^{11} + 2777274333782403846t^{10} + 316289952497869874t^{9} + 25990031617860993t^{8} + 1698917909290478t^{7} + 136599749899323t^{6} + 14108727184141t^{5} + 982233332272t^{4} + 35607428602t^{3} + 1029449108t^{2} + 1404495t + 20286 \right)D^{5} + \left( 1026738552475882003968000t^{25} + 3809637651814448556134400t^{24} + 7053921174686321223402240t^{23} + 7939242679694689607721216t^{22} + 5957991717621208500999936t^{21} + 3110189973363221116003392t^{20} + 1154605519562419756576224t^{19} + 339887181162681550406424t^{18} + 125425133168191511294376t^{17} + 72762999359749759715388t^{16} + 39672682207025087726740t^{15} + 16059220846105662730484t^{14} + 4737881213001892010280t^{13} + 1030989868347582279292t^{12} + 166612653206891026952t^{11} + 19983602359880165118t^{10} + 1767913161023197694t^{9} + 112773882460765501t^{8} + 4428057941494509t^{7} - 41282627670373t^{6} - 21262386502988t^{5} - 1379443087881t^{4} - 39427380206t^{3} - 878679250t^{2} - 809439t - 10143 \right)D^{4} + t \left( 4312301920398704416665600t^{24} + 15432245127754798128230400t^{23} + 27653135239954709847206400t^{22} + 29724211309172954360676096t^{21} + 21067113798281986192027968t^{20} + 10258129822037106445433696t^{19} + 3535087294280902621983632t^{18} + 1059012971451847047337272t^{17} + 495297362430941097715304t^{16} + 317550153863266354478036t^{15} + 164171805926673250386360t^{14} + 60954267617803060382972t^{13} + 16397985594762465190660t^{12} + 3244583383532141575848t^{11} + 475333216781420503628t^{10} + 51568264722034110922t^{9} + 4137330143597687598t^{8} + 246354472603212667t^{7} + 10933517220959018t^{6} + 344746438893345t^{5} + 6337851859799t^{4} + 15530528388t^{3} - 713773518t^{2} + 13757818t + 15111 \right)D^{3} + 2 t ^{2} \left( 4764066883488092498411520t^{23} + 16568664335180563975772160t^{22} + 28936431511179985444775040t^{21} + 29959943499597218732599680t^{20} + 20287281936039951642931680t^{19} + 9356237204308998320572832t^{18} + 3069403940029878581411200t^{17} + 985712451123346172173296t^{16} + 555657445962846774052592t^{15} + 362758285783606434663124t^{14} + 176779486343274684386400t^{13} + 61089137413680902190292t^{12} + 15252685463343048687704t^{11} + 2794109839259763440544t^{10} + 378130327251757941736t^{9} + 37897137711158219990t^{8} + 2821947312227551788t^{7} + 156993386748712109t^{6} + 6413107721353894t^{5} + 170330416556001t^{4} + 2273130081855t^{3} + 9009088428t^{2} + 498867471t + 5913369 \right)D^{2} + 24 t ^{2} \left( 431230192039870441666560t^{23} + 1467890591012957073187840t^{22} + 2514698661038135378856000t^{21} + 2529189448342341347155008t^{20} + 1654118220975441929747968t^{19} + 732469833062208812524976t^{18} + 233306229515571730701800t^{17} + 81642548352969543903580t^{16} + 51889004691909952615456t^{15} + 33537291903710736464580t^{14} + 15566440896006446958328t^{13} + 5092223247384151449828t^{12} + 1200757377440174357080t^{11} + 207221577140481991366t^{10} + 26368820593442747322t^{9} + 2487275721979163485t^{8} + 175246386078002300t^{7} + 9249747197383439t^{6} + 347625602134565t^{5} + 7330217133492t^{4} + 38531714344t^{3} - 227364776t^{2} + 29085501t + 223146 \right)D + 144 t ^{2} \left( 29335387213596628684800t^{23} + 98392286623572925491200t^{22} + 166332923179146212211840t^{21} + 163884904820558868877440t^{20} + 104587084214770618096560t^{19} + 45014032578186786363120t^{18} + 14105947151559296040776t^{17} + 5318723462391059827656t^{16} + 3624387673957228400088t^{15} + 2309251907747412455880t^{14} + 1035378616015570788876t^{13} + 325934932738402522528t^{12} + 73788481707856196348t^{11} + 12194778568066107926t^{10} + 1483265390500036182t^{9} + 133854918009104940t^{8} + 9064720702989480t^{7} + 459752422448560t^{6} + 16077218723044t^{5} + 270448084704t^{4} - 732358315t^{3} - 32844977t^{2} + 843525t + 6762 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 8)

|λ| = 14.86381 λ ≈ 14.86381; multiplicity 1; Jordan blocks 1 λ ≈ 1.531571 - 7.299112i; multiplicity 1; Jordan blocks 1 λ ≈ 1.531571 + 7.299112i; multiplicity 1; Jordan blocks 1 λ ≈ -2.963477 - 2.305931i; multiplicity 1; Jordan blocks 1 λ ≈ -2.963477 + 2.305931i; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2 + 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 2 2
Sector dimension
10
Spectral radius
$14.863812$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14.86381$11
$1.531571 - 7.299112\mathrm{i}$11
$1.531571 + 7.299112\mathrm{i}$11
$-2.963477 - 2.305931\mathrm{i}$11
$-2.963477 + 2.305931\mathrm{i}$11
$-2 - 2\mathrm{i}$21, 1
$-2 + 2\mathrm{i}$21, 1
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{2} + 4 \lambda + 8\right)^{2} \left(\lambda^{5} - 12 \lambda^{4} + 9 \lambda^{3} - 480 \lambda^{2} - 3474 \lambda - 11657\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{5} - 12 \lambda^{4} + 9 \lambda^{3} - 480 \lambda^{2} - 3474 \lambda - 11657\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 12 & 36 & 18 & 30 & 78 & 136 & 144 & 570 \\ 1 & 0 & 3 & 6 & -6 & 12 & 18 & -6 & -18 & -24 \\ 1 & 0 & 1 & 4 & 4 & 4 & 12 & 30 & 36 & 144 \\ 0 & 1 & 1 & 1 & -1 & 2 & 4 & 0 & -8 & -18 \\ 0 & 1 & 0 & 3 & 1 & 0 & 3 & 12 & 12 & 36 \\ 0 & 0 & 1 & 0 & 1 & -4 & -4 & -2 & -4 & -18 \\ 0 & 0 & 1 & 0 & 0 & 2 & 2 & 4 & 8 & 36 \\ 0 & 0 & 0 & 3 & 0 & 4 & 8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & -1 & -\frac{3}{2} & 2 & 1 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{4}{3} & 1 & 0\end{pmatrix}\notag\end{equation}$$