Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-240-54-a

·

Blow-up of a smooth quadric fourfold along a del Pezzo surface of degree $5$.

  • $\pi_1\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a quintic del Pezzo surface.

  • $\pi_2\colon X\to\mathbb P^3$ is given on $Q^4$ (1-512-105-a) by the four quadrics through the surface; its fibers are lines, except over $5$ points, where they are planes.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3ω1ω2ω3
Bundle
E^(w1 + w2)
Bundle
O(1,1)
ZL62 label
32.L
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 240
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 240
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 101
$\mathrm{c}_1\mathrm{c}_3$ = 46
$\mathrm{c}_4$ = 13
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 7 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 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 46 0 0 x1 - 10 56
$\mathrm{HH}^{ 4 }$ = 54 0 0 0 0 54
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 49t + 140t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(124)
Coates et al. (2020) Str(2)
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( t + 1 \right) \left( 4339t^{8} + 7120t^{7} + 7833t^{6} + 7224t^{5} + 2213t^{4} + 661t^{3} + 15t^{2} - 2t - 1 \right) \left( 28167035939432t^{13} + 40325518260328t^{12} + 43112101764912t^{11} + 50872979628368t^{10} + 16754587415262t^{9} - 13962720816712t^{8} - 8026310122222t^{7} - 970510751930t^{6} + 46860780026t^{5} + 58315464699t^{4} + 789308204t^{3} + 1329858457t^{2} + 11438296t + 196320 \right)D^{6} + 2 \left( 1283276073882552204t^{22} + 4668208212101781060t^{21} + 9047721208237471924t^{20} + 13471471349925984196t^{19} + 15273613540860273305t^{18} + 12066333520087885761t^{17} + 6006648342541221344t^{16} + 721736868937722152t^{15} - 1754080235519965035t^{14} - 1580038204986216445t^{13} - 632581752949676905t^{12} - 128759939233472067t^{11} - 7506134582359137t^{10} + 3518281301742021t^{9} + 1045450770211745t^{8} + 191349026157098t^{7} + 22836547073788t^{6} + 4180640756030t^{5} + 249451538477t^{4} + 3008576730t^{3} + 2650215950t^{2} + 16568484t + 196320 \right)D^{5} + \left( 21387934564709203400t^{22} + 70362857964441404600t^{21} + 125525112991732355216t^{20} + 177969219745469529376t^{19} + 188713365148325377006t^{18} + 124977623085417741478t^{17} + 35597105362922070240t^{16} - 19354073109782886494t^{15} - 29361088368914300504t^{14} - 16478348780031106656t^{13} - 4721691993258795495t^{12} - 506335384768647306t^{11} + 99412437666434149t^{10} + 51060700756862466t^{9} + 10924933180665440t^{8} + 2095930371063813t^{7} + 255866389181406t^{6} + 25493769690324t^{5} - 164686066749t^{4} + 9311281787t^{3} - 5148624569t^{2} - 35320008t - 196320 \right)D^{4} + t \left( 89829325171778654280t^{21} + 270969983067712576200t^{20} + 453664696943334647640t^{19} + 630595761699387018360t^{18} + 633183743776865858430t^{17} + 340838672192979009142t^{16} + 11796961018877102640t^{15} - 122133014020310537224t^{14} - 98536987797626816826t^{13} - 39419242871611963244t^{12} - 7619038362882375358t^{11} + 154334124413079634t^{10} + 520432927606767215t^{9} + 153084140199610018t^{8} + 26329159521219787t^{7} + 4853339499444878t^{6} + 576575903707545t^{5} + 45510744669364t^{4} + 795714033629t^{3} + 15419272836t^{2} + 133496798t + 75256 \right)D^{3} + t ^{2} \left( 198480032760501407552t^{20} + 557206642416545074520t^{19} + 890279275626125048544t^{18} + 1234909671353344433072t^{17} + 1180113584060041313344t^{16} + 499978214942700858912t^{15} - 134282450518712391972t^{14} - 283536856577556828480t^{13} - 163129725261866248448t^{12} - 47068780535236488171t^{11} - 4312509275724259400t^{10} + 1813917878295906866t^{9} + 969542350598350586t^{8} + 234015765967589746t^{7} + 39085384205100850t^{6} + 7323074995558190t^{5} + 771843030704618t^{4} + 45015759151085t^{3} + 792459974766t^{2} + 18097155420t + 104179680 \right)D^{2} + 2 t ^{2} \left( 107795190206134385136t^{20} + 286100765435666622840t^{19} + 442267023288866760656t^{18} + 617301403299155363984t^{17} + 564354458495379068500t^{16} + 183399856682912172292t^{15} - 119345891708007145516t^{14} - 147498152377086736856t^{13} - 65953739631297691916t^{12} - 13000211458905944633t^{11} + 750033949423408808t^{10} + 1167847344378753692t^{9} + 427866441053897944t^{8} + 92813894968562582t^{7} + 16016693075321888t^{6} + 2961158277896292t^{5} + 270438058717268t^{4} + 12014351188551t^{3} + 239433315272t^{2} + 5209262816t + 20417280 \right)D + 8 t ^{2} \left( 10999509204707590320t^{20} + 28055086436712673800t^{19} + 42427432763497541328t^{18} + 59690095241414678088t^{17} + 52679954763540581668t^{16} + 13101557680190355172t^{15} - 14695850260288319924t^{14} - 14174138143240224232t^{13} - 5183261821401148552t^{12} - 607112936132568899t^{11} + 219687513857377376t^{10} + 119579020856522899t^{9} + 36715862695493436t^{8} + 7463610820590630t^{7} + 1354990600889970t^{6} + 242237313683779t^{5} + 19317671247398t^{4} + 682238068991t^{3} + 16041271320t^{2} + 306998672t + 785280 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 4)

|λ| = 9.757177 λ ≈ 9.757177; multiplicity 1; Jordan blocks 1 λ ≈ -4.215126 - 6.483435i; multiplicity 1; Jordan blocks 1 λ ≈ -4.215126 + 6.483435i; multiplicity 1; Jordan blocks 1 λ ≈ -1.087244 - 2.638216i; multiplicity 1; Jordan blocks 1 λ ≈ -1.087244 + 2.638216i; multiplicity 1; Jordan blocks 1 λ ≈ -0.1199081 - 0.9931465i; multiplicity 1; Jordan blocks 1 λ ≈ -0.1199081 + 0.9931465i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ -0.9126206; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$9.75717659$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.757177$11
$-4.215126 - 6.483435\mathrm{i}$11
$-4.215126 + 6.483435\mathrm{i}$11
$-1.087244 - 2.638216\mathrm{i}$11
$-1.087244 + 2.638216\mathrm{i}$11
$-0.1199081 - 0.9931465\mathrm{i}$11
$-0.1199081 + 0.9931465\mathrm{i}$11
$-1$11
$-0.9126206$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{8} + 2 \lambda^{7} - 15 \lambda^{6} - 661 \lambda^{5} - 2213 \lambda^{4} - 7224 \lambda^{3} - 7833 \lambda^{2} - 7120 \lambda - 4339\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{8} + 2 \lambda^{7} - 15 \lambda^{6} - 661 \lambda^{5} - 2213 \lambda^{4} - 7224 \lambda^{3} - 7833 \lambda^{2} - 7120 \lambda - 4339\right)\notag\end{equation}$$

Quantum multiplication matrix

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