Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-178-42-a

·

Blow-up of a hyperplane section of the adjoint $G_2$ fivefold along a ruled surface over a conic.

  • $\pi_1\colon X\to D$ is the blow-up of the Mukai fourfold $D$ of genus $10$, a hyperplane section of the adjoint $G_2$-variety (1-288-63-a), along a $\mathbb P^1$-bundle over a conic.

  • $\pi_2\colon X\to Q^3$, onto a smooth quadric threefold, is a conic bundle whose discriminant is a quadric section of $Q^3$.

Varieties in this family can be blown down to

Defining data
Ambient
Gr(2, 4) × Gr(2, 7)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w6) + E^(w1 + w4) + E^(w2)
Bundle
Q_2^v(1) + U_1^v box U_2^v + O_1(1)
ZL62 label
31601.51w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 42
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 178
$\chi(X, \mathrm{T}_X)$ = -3

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 178
$\mathrm{c}_1^2\mathrm{c}_2$ = 136
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 3 0 x0 + 3 x1
$\mathrm{HH}^{ 3 }$ = x1 + 36 0 0 x1 + 9 27
$\mathrm{HH}^{ 4 }$ = 42 0 0 0 0 42
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 37t + 102t^{2} + 37t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( 5t + 1 \right) \left( t + 1 \right) ^{2} \left( 4432t^{6} - 96t^{5} - 1064t^{4} - 608t^{3} - 91t^{2} - 2t + 1 \right) \left( 2844535302012t^{11} + 5067752130764t^{10} + 4802167843987t^{9} + 3646546162791t^{8} + 2147034555517t^{7} + 787034460214t^{6} + 150049478192t^{5} + 11548999304t^{4} - 42650417t^{3} - 54818252t^{2} - 695741t + 28569 \right)D^{6} + \left( t + 1 \right) \left( 1323732948144304320t^{19} + 3544226410616642368t^{18} + 4404727324154729008t^{17} + 3641167229028317904t^{16} + 2250441097980592164t^{15} + 845589964479096860t^{14} - 55148268357055195t^{13} - 278465094774517117t^{12} - 168658678626084257t^{11} - 55250781524466089t^{10} - 10980125300803476t^{9} - 1310774213405988t^{8} - 93320616629533t^{7} - 6347301595383t^{6} - 792710049116t^{5} - 60327106348t^{4} + 269244306t^{3} + 258262348t^{2} + 2982947t - 85707 \right)D^{5} + \left( 11031107901202536000t^{20} + 38073599037110709760t^{19} + 60363401741379490752t^{18} + 62419775857334967696t^{17} + 49263024049440443712t^{16} + 29591706965664379316t^{15} + 11494077732976845377t^{14} + 1473685892729541960t^{13} - 1035783856169659305t^{12} - 661620393314413269t^{11} - 182778057663937861t^{10} - 27291690562217390t^{9} - 1738877062431738t^{8} + 112647683744782t^{7} + 35762989075947t^{6} + 3618943213596t^{5} + 168272329213t^{4} + 1536369795t^{3} - 257207291t^{2} - 1902782t + 57138 \right)D^{4} + t \left( 46330653185050651200t^{19} + 151683110412404289280t^{18} + 234357561394699994480t^{17} + 244366403670146606688t^{16} + 199451194157871873140t^{15} + 125467573051836442040t^{14} + 53827541815304171375t^{13} + 12397102569841354120t^{12} - 173921750668019724t^{11} - 994727909042964491t^{10} - 286754646405704740t^{9} - 37366821128397075t^{8} - 1591190196609090t^{7} + 171079447902598t^{6} + 23583660845905t^{5} + 374416261220t^{4} - 63444011572t^{3} - 2713611953t^{2} - 42325910t + 96621 \right)D^{3} + t \left( 102368681323159534080t^{19} + 321241577444890688896t^{18} + 485697399365838087216t^{17} + 507543265717837423120t^{16} + 421127226137716729452t^{15} + 269766540341552767636t^{14} + 119636521148577122532t^{13} + 31661087519139708902t^{12} + 3135810732171540719t^{11} - 686102412137239925t^{10} - 260727282798946237t^{9} - 32430922921877383t^{8} - 1126242576757754t^{7} + 123706751969826t^{6} + 8891039835532t^{5} - 989622770558t^{4} - 103897538651t^{3} - 2745227667t^{2} - 20923633t + 28569 \right)D^{2} + 2 t ^{2} \left( 55596783822060781440t^{18} + 168932905493924955776t^{17} + 251077862645391054832t^{16} + 262267362655584090928t^{15} + 219466989902443962956t^{14} + 141573229002449110028t^{13} + 63445798170359446286t^{12} + 17621551623213139606t^{11} + 2414592219607664045t^{10} - 42818818080964305t^{9} - 65015158536711156t^{8} - 8992135723532906t^{7} - 470294084835780t^{6} - 9443481117654t^{5} - 3247370753646t^{4} - 635327531248t^{3} - 35909829227t^{2} - 792686763t - 2111792 \right)D + 24 t ^{3} \left( 1891047068777577600t^{17} + 5618837145070720640t^{16} + 8249178359859704752t^{15} + 8607404111309636848t^{14} + 7235083574452391772t^{13} + 4678584926941433116t^{12} + 2101299235507907818t^{11} + 594198011826671182t^{10} + 90679176727608925t^{9} + 3216677542015571t^{8} - 1161627629043610t^{7} - 206308575181416t^{6} - 17525038954952t^{5} - 1548501748490t^{4} - 204659288472t^{3} - 19694800118t^{2} - 809213283t - 16198623 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 7)

|λ| = 13.0334 λ ≈ 13.0334; multiplicity 1; Jordan blocks 1 λ ≈ -4.39225 - 3.338627i; multiplicity 1; Jordan blocks 1 λ ≈ -4.39225 + 3.338627i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 λ ≈ -1.83674 - 2.113893i; multiplicity 1; Jordan blocks 1 λ ≈ -1.83674 + 2.113893i; multiplicity 1; Jordan blocks 1 λ ≈ 1.424578; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
9
Spectral radius
$13.0334022$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.0334$11
$-4.39225 - 3.338627\mathrm{i}$11
$-4.39225 + 3.338627\mathrm{i}$11
$-5$11
$-1.83674 - 2.113893\mathrm{i}$11
$-1.83674 + 2.113893\mathrm{i}$11
$1.424578$11
$-1$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 5\right) \left(\lambda^{6} - 2 \lambda^{5} - 91 \lambda^{4} - 608 \lambda^{3} - 1064 \lambda^{2} - 96 \lambda + 4432\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 5\right) \left(\lambda^{6} - 2 \lambda^{5} - 91 \lambda^{4} - 608 \lambda^{3} - 1064 \lambda^{2} - 96 \lambda + 4432\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 10 & 6 & 36 & 18 & 40 & 8 & 40 \\ 1 & -1 & 2 & 2 & 6 & 4 & 18 & 6 & 16 \\ 1 & 1 & 0 & 2 & 10 & 4 & 6 & 0 & 8 \\ 0 & 2 & 0 & -1 & 4 & 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 & -1 & 1 & 4 & 2 & 6 \\ 0 & 0 & 3 & 1 & 3 & -1 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 8 & 5 & 2 & 1 & 6 \\ 0 & 0 & 0 & 1 & -9 & -9 & -4 & -3 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 1 & 0\end{pmatrix}\notag\end{equation}$$