Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-116-30-c

·

Blow-up of a smooth quadric fourfold along a determinantal surface.

  • $\pi_1\colon X\to Q$ is the blow-up $\mathrm{Bl}_SQ$ of the smooth quadric fourfold $Q=\mathrm{Gr}(2,4)$ (1-512-105-a) along a smooth rational surface $S$ of degree $9$, isomorphic to the blow-up of a del Pezzo surface of degree $4$ at seven points (so to the blow-up of $\mathbb P^2$ at $12$ points).

  • $\pi_2\colon X\to Y$ is the blow-up $\mathrm{Bl}_{S}Y$ of a smooth intersection $Y\subset\mathbb P^6$ of two quadrics (1-324-70-a) containing a plane, along a smooth cubic surface $S$ embedded with degree $7$, a conic bundle over $\mathbb P^1$ with $5$ singular fibers (the intersection of $Y$ with a rational normal threefold scroll of degree $4$).

Varieties in this family can be blown down to

Defining data
Ambient
ℙ6P6 × Gr(2, 4)
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8 (marked)node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
2 copies of E^(w1 + w7) + E^(w1 + w9)
Bundle
2 copies of O_1(1) box U_2^v + O_1(1) box Q_2
ZL62 label
31600.wFw2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7;4) \\ \det\mathcal{E} &\cong \mathcal{O}(6;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 116
$\chi(X, \mathrm{T}_X)$ = -16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 116
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 121
$\mathrm{c}_1\mathrm{c}_3$ = 30
$\mathrm{c}_4$ = 21
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 15 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 + 16 0 x0 + 16 x1
$\mathrm{HH}^{ 3 }$ = x1 + 26 0 0 x1 + 14 12
$\mathrm{HH}^{ 4 }$ = 30 0 0 0 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 64t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(547)
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( t + 1 \right) \left( 2t + 1 \right) ^{2} \left( 1366t^{7} + 4899t^{6} + 7290t^{5} + 5777t^{4} + 2326t^{3} + 243t^{2} - 3t - 1 \right) \left( 408062635776t^{17} - 17060766019776t^{16} - 100124140657584t^{15} - 234815599063392t^{14} - 298306387804676t^{13} - 213170800933000t^{12} - 63132646609803t^{11} + 28107102117386t^{10} + 38332984879009t^{9} + 18418313012382t^{8} + 4681926309525t^{7} + 519246884256t^{6} - 37531307370t^{5} - 18497426299t^{4} - 2487795782t^{3} - 269728341t^{2} - 2689277t - 13804 \right)D^{6} + \left( 2t + 1 \right) \left( 23411369539740672t^{26} - 931252482519860736t^{25} - 10313226385949878464t^{24} - 48055198797726064512t^{23} - 133531679997010987744t^{22} - 249243959550956740592t^{21} - 328309903757564104408t^{20} - 307162213752159600300t^{19} - 193345608739437858143t^{18} - 60791249333513900551t^{17} + 22057615194318989638t^{16} + 42802955442462586914t^{15} + 30189545372518349717t^{14} + 13557889124850420942t^{13} + 4147501547937178084t^{12} + 822577475901870652t^{11} + 76470786575691283t^{10} - 9290014065393095t^{9} - 4693967343808403t^{8} - 890721820321666t^{7} - 112012657297863t^{6} - 9612049054523t^{5} - 429488024267t^{4} - 13762044278t^{3} - 1060036926t^{2} - 7791751t - 27608 \right)D^{5} + \left( 390189492329011200t^{27} - 16279398903231590400t^{26} - 176958435443565334656t^{25} - 812322106330625494848t^{24} - 2223856987933794453440t^{23} - 4085357397325195649344t^{22} - 5281447310237759433328t^{21} - 4812108540777205512056t^{20} - 2871475860372047914586t^{19} - 702413804294900521163t^{18} + 596348920615658876113t^{17} + 860643169078252707068t^{16} + 586932910296203823324t^{15} + 266400047731421402171t^{14} + 84514060717167566806t^{13} + 17867775068256094454t^{12} + 1886608034583651053t^{11} - 199029681511035774t^{10} - 128494930143734693t^{9} - 29051697883361377t^{8} - 4332066650131088t^{7} - 453380937223362t^{6} - 27937102052736t^{5} - 622530633924t^{4} + 235998188t^{3} + 983335023t^{2} + 6944951t + 13804 \right)D^{4} + t \left( 1638795867781847040t^{26} - 71520417897987456000t^{25} - 731340269408597827200t^{24} - 3160993949072866991424t^{23} - 8108138005297551578048t^{22} - 13841028112542940505344t^{21} - 16377892557785361547200t^{20} - 13201038318638195491584t^{19} - 6200330511513090969446t^{18} + 158144859765213429717t^{17} + 3051611365083855679075t^{16} + 2908602532075882981210t^{15} + 1659285919085548086396t^{14} + 651533540916479681735t^{13} + 176478772538352469914t^{12} + 29318671434728824992t^{11} + 1007203562577085252t^{10} - 964705925740727515t^{9} - 320376237062500989t^{8} - 61128159550073311t^{7} - 8435602483167682t^{6} - 808380333859263t^{5} - 41779719680871t^{4} - 811869435707t^{3} - 11012799546t^{2} - 69076972t - 20349 \right)D^{3} + 2 t ^{2} \left( 1810479244406611968t^{25} - 81672948794603393280t^{24} - 796044196330715152896t^{23} - 3278514008235606496896t^{22} - 7979251313354532803968t^{21} - 12828549509269665644400t^{20} - 14086503118833142640896t^{19} - 10140552418600531676312t^{18} - 3550498454848208479180t^{17} + 1545651361364657529141t^{16} + 3245823124918800576409t^{15} + 2547213944991194021885t^{14} + 1285480217885708014335t^{13} + 449069607216576492498t^{12} + 105109525767709029383t^{11} + 13045376777374748636t^{10} - 894479915179726531t^{9} - 837605570681341451t^{8} - 218587958922424986t^{7} - 38506805629848730t^{6} - 5088829513948004t^{5} - 445097378134347t^{4} - 19087395197585t^{3} - 326454238838t^{2} - 5020951327t - 27115464 \right)D^{2} + 4 t ^{2} \left( 983277520669108224t^{25} - 45415500458213318400t^{24} - 427078044265604383104t^{23} - 1695311545768237617984t^{22} - 3962255673215934100928t^{21} - 6077571421936915061328t^{20} - 6278802812811118737056t^{19} - 4083871055509814613992t^{18} - 972938560895585908038t^{17} + 1139360803984495140862t^{16} + 1620186775281550977846t^{15} + 1128053042754459782712t^{14} + 519393539953314451492t^{13} + 164520066088537071844t^{12} + 33480797511989167044t^{11} + 2757440127876977045t^{10} - 734403641581989329t^{9} - 344930169802279149t^{8} - 79260796270531489t^{7} - 13395077129890329t^{6} - 1705387271501070t^{5} - 135330752090320t^{4} - 4882295941967t^{3} - 81649029054t^{2} - 1161606852t - 5245520 \right)D + 8 t ^{2} \left( 200668881769205760t^{25} - 9414431564033088000t^{24} - 86390770182372695808t^{23} - 334216941217987185792t^{22} - 758975468476556373888t^{21} - 1125072585122530372704t^{20} - 1109928489051293067264t^{19} - 663448121060544127056t^{18} - 92885765276708822316t^{17} + 257156089446352330604t^{16} + 305736963418099595398t^{15} + 196753030756126203357t^{14} + 84608425097892257351t^{13} + 24737166833363766745t^{12} + 4412745234971593203t^{11} + 177923222646011240t^{10} - 154866231037230659t^{9} - 55723735423287213t^{8} - 11890592923865528t^{7} - 1965495723579985t^{6} - 241702690675788t^{5} - 17462707547126t^{4} - 543300574171t^{3} - 9495735798t^{2} - 111917834t - 386512 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ6P6 × Gr(2, 4)

|λ| = 18.27528 λ ≈ 18.27528; multiplicity 1; Jordan blocks 1 λ ≈ -9.140698 - 1.843384i; multiplicity 1; Jordan blocks 1 λ ≈ -9.140698 + 1.843384i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ -0.6647248 - 0.8704433i; multiplicity 1; Jordan blocks 1 λ ≈ -0.6647248 + 0.8704433i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ -0.8322185 - 0.1551015i; multiplicity 1; Jordan blocks 1 λ ≈ -0.8322185 + 0.1551015i; multiplicity 1; Jordan blocks 1 2
Sector dimension
10
Spectral radius
$18.2752829$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.27528$11
$-9.140698 - 1.843384\mathrm{i}$11
$-9.140698 + 1.843384\mathrm{i}$11
$-2$22
$-0.6647248 - 0.8704433\mathrm{i}$11
$-0.6647248 + 0.8704433\mathrm{i}$11
$-1$11
$-0.8322185 - 0.1551015\mathrm{i}$11
$-0.8322185 + 0.1551015\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right)^{2} \left(\lambda^{7} + 3 \lambda^{6} - 243 \lambda^{5} - 2326 \lambda^{4} - 5777 \lambda^{3} - 7290 \lambda^{2} - 4899 \lambda - 1366\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right)^{2} \left(\lambda^{7} + 3 \lambda^{6} - 243 \lambda^{5} - 2326 \lambda^{4} - 5777 \lambda^{3} - 7290 \lambda^{2} - 4899 \lambda - 1366\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 14 & 14 & 78 & 78 & 30 & 24 & 232 & 200 & 390 \\ 1 & -1 & 2 & 24 & 26 & 8 & 6 & 39 & 33 & 96 \\ 1 & 3 & -1 & 16 & 8 & 10 & 8 & 42 & 39 & 72 \\ 0 & 1 & 1 & -2 & 0 & -1 & 0 & 10 & 8 & 9 \\ 0 & 1 & 0 & 2 & -1 & 2 & 1 & 6 & 6 & 9 \\ 0 & 0 & 1 & 3 & 5 & 0 & 0 & 10 & 8 & 18 \\ 0 & 0 & 1 & 3 & 4 & 0 & -1 & 0 & 0 & 9 \\ 0 & 0 & 0 & -3 & -\frac{32}{3} & \frac{8}{3} & \frac{5}{3} & -\frac{16}{3} & -\frac{5}{3} & -14 \\ 0 & 0 & 0 & 8 & \frac{49}{3} & -\frac{4}{3} & -\frac{1}{3} & \frac{23}{3} & \frac{10}{3} & 28 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{7}{3} & 0\end{pmatrix}\notag\end{equation}$$