Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-272-60-a

·

Blow-up of a smooth quadric fourfold along a balanced rational normal quartic scroll.

  • $\pi_1\colon X\to Q^3$, onto a smooth quadric threefold, is a fibration with general fiber $\mathbb P^1$ and fiber $\mathbb P^2$ over exactly two points.

  • $\pi_2\colon X\to Q^4$ is the blow-up of a smooth quadric fourfold $Q^4\subset\mathbb P^5$ (1-512-105-a) along a rational normal quartic scroll $\mathbb P^1\times\mathbb P^1$, embedded by $\mathcal O(2,1)$.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ5P5 × 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_7 (marked)node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w7) + 2 copies of E^(w1 + w6)
Bundle
O_2(1) + 2 copies of O_1(1) box U_2^v
ZL62 label
3150.1w2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6;4) \\ \det\mathcal{E} &\cong \mathcal{O}(4;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 60
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 272
$\chi(X, \mathrm{T}_X)$ = 1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 272
$\mathrm{c}_1^2\mathrm{c}_2$ = 164
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 10
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 4 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 - 1 0 x0 - 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 50 0 0 x1 - 17 67
$\mathrm{HH}^{ 4 }$ = 60 0 0 0 0 60
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 55t + 160t^{2} + 55t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(115)
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( 3t + 1 \right) \left( 9t - 1 \right) \left( 8t^{2} + 4t + 1 \right) \left( 8t^{3} + 52t^{2} + 7t + 1 \right) \left( 160721210880t^{13} + 96204837888t^{12} - 49462016592t^{11} + 322030724216t^{10} + 597190179636t^{9} + 555596343760t^{8} + 251334512038t^{7} + 29077864036t^{6} - 8422520716t^{5} - 2809642279t^{4} - 196961263t^{3} - 23742143t^{2} - 145219t - 4114 \right)D^{6} + \left( 4721346290810880t^{21} + 25457323249631232t^{20} + 9223999773281280t^{19} - 4414025331394560t^{18} + 74254988941194240t^{17} + 122475384240078720t^{16} + 82069395118734672t^{15} - 5331612463767040t^{14} - 48171333597697980t^{13} - 31250748460410368t^{12} - 8441109564405454t^{11} - 343350823214520t^{10} + 402135246627060t^{9} + 117066470139930t^{8} + 15550705237109t^{7} + 1552942007844t^{6} + 197580457191t^{5} + 19484795726t^{4} + 1014611985t^{3} + 94634536t^{2} + 402745t + 8228 \right)D^{5} + \left( 31938519026073600t^{21} + 137264019351994368t^{20} + 53269791491349504t^{19} + 13761480916270080t^{18} + 504363563450204352t^{17} + 850874192340613024t^{16} + 708975470452818288t^{15} + 191136466654405912t^{14} - 137789198436114132t^{13} - 108892067048046344t^{12} - 24130803673007462t^{11} + 2112055021670552t^{10} + 2114924621958910t^{9} + 490740884559865t^{8} + 67492648511909t^{7} + 6649681653101t^{6} + 299496887711t^{5} - 17234749319t^{4} - 783175431t^{3} - 92793461t^{2} - 472785t - 4114 \right)D^{4} + t \left( 109701869698252800t^{20} + 380924929032192000t^{19} + 142268390598878208t^{18} + 114322266722967552t^{17} + 1634680404099426816t^{16} + 2868702341698213248t^{15} + 2724015440284734960t^{14} + 1106763099517451904t^{13} - 134534921452309028t^{12} - 214777542869745808t^{11} - 43786703895386290t^{10} + 8088200720211736t^{9} + 4750061035300080t^{8} + 927639596111328t^{7} + 118602951228911t^{6} + 12125985099856t^{5} + 641824273689t^{4} + 10593696976t^{3} + 242949707t^{2} + 2478312t + 1331 \right)D^{3} + 2 t ^{2} \left( 100536903369031680t^{19} + 287459470332592128t^{18} + 93961316181602304t^{17} + 139291159768899840t^{16} + 1391887846427244960t^{15} + 2516680852444108976t^{14} + 2606813535538647792t^{13} + 1242550459099597960t^{12} + 40839376806120664t^{11} - 131001989188827988t^{10} - 29941533589236836t^{9} + 5274590679212586t^{8} + 2778544728566885t^{7} + 514236870539554t^{6} + 69267933091578t^{5} + 7087881274966t^{4} + 313591221201t^{3} + 5456443970t^{2} + 146520772t + 969408 \right)D^{2} + 8 t ^{2} \left( 23190142075453440t^{19} + 55925940340555776t^{18} + 14417395205239296t^{17} + 37524058979754240t^{16} + 299417444872279200t^{15} + 550218898291220304t^{14} + 605971019396850408t^{13} + 314631952334128352t^{12} + 29784390634088662t^{11} - 23402185506849732t^{10} - 6701669452120102t^{9} + 676331762602458t^{8} + 401024398683302t^{7} + 76263914638686t^{6} + 11357941132927t^{5} + 1126298555590t^{4} + 41715491254t^{3} + 846401678t^{2} + 21652421t + 98736 \right)D + 16 t ^{2} \left( 4165893786009600t^{19} + 8732031160614912t^{18} + 1568370986809344t^{17} + 7437739209911424t^{16} + 50788549839600144t^{15} + 93891159109610712t^{14} + 108018200082170256t^{13} + 59095706516743512t^{12} + 7340379117284772t^{11} - 3665000919679630t^{10} - 1287347245656115t^{9} + 43384168524452t^{8} + 44229224656021t^{7} + 9395741780792t^{6} + 1588486215922t^{5} + 150021729136t^{4} + 4754672289t^{3} + 117013158t^{2} + 2626063t + 8228 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ5P5 × Gr(2, 4)

|λ| = 9 λ ≈ 9; multiplicity 1; Jordan blocks 1 λ ≈ -3.421453 - 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -3.421453 + 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ 1; multiplicity 1; Jordan blocks 1 λ ≈ -0.1570937; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$9$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9$11
$-3.421453 - 6.262482\mathrm{i}$11
$-3.421453 + 6.262482\mathrm{i}$11
$-3$11
$-2 - 2\mathrm{i}$11
$-2 + 2\mathrm{i}$11
$1$11
$-0.1570937$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 3\right) \left(\lambda - 1\right) \left(\lambda - 9\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{3} + 7 \lambda^{2} + 52 \lambda + 8\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 3\right) \left(\lambda - 1\right) \left(\lambda - 9\right) \left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{3} + 7 \lambda^{2} + 52 \lambda + 8\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 12 & 12 & 3 & 8 & 16 & 0 \\ 2 & 0 & 2 & 0 & 0 & 0 & 3 & 12 & 4 \\ 1 & 0 & -1 & 2 & 4 & 0 & 6 & 12 & 0 \\ 0 & 1 & 2 & -1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 2 & 0 & 2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & -1 & 2 & 8 & 0 \\ 0 & 0 & 0 & 2 & -4 & 3 & -3 & -4 & 2 \\ 0 & 0 & 0 & 2 & 4 & -\frac{1}{4} & \frac{3}{2} & 2 & -\frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 12 & 0\end{pmatrix}\notag\end{equation}$$