Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-188-44-a

·

Blow-up of a hyperplane section of the adjoint $G_2$ fivefold along a del Pezzo surface of degree $6$.

  • $\pi_1\colon X\to Y$ is the blow-up of a smooth hyperplane section $Y$ of the adjoint $G_2$-variety $G_2/P_2\subset\mathbb P^{13}$ (1-288-63-a) along a sextic del Pezzo surface.

  • $\pi_2\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a sextic del Pezzo surface.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1) + E^(w2) + E^(w3)
Bundle
O(1,0) + O(0,1) + Q_1^v ot U_2^v ot U_3^v
ZL62 label
602.210
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,6) \\ \det\mathcal{E} &\cong \mathcal{O}(1,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 44
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 188
$\chi(X, \mathrm{T}_X)$ = -5

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 188
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 6 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 + 5 0 x0 + 5 x1
$\mathrm{HH}^{ 3 }$ = x1 + 38 0 0 x1 + 5 33
$\mathrm{HH}^{ 4 }$ = 44 0 0 0 0 44
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 39t + 108t^{2} + 39t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( t + 1 \right) ^{2} \left( 4768t^{5} + 2512t^{4} + 912t^{3} + 80t^{2} - 2t - 1 \right) \left( 814648512t^{7} + 981015360t^{6} + 602617472t^{5} + 218707888t^{4} - 27008t^{3} + 3339564t^{2} + 12573t + 329 \right)D^{6} + \left( t + 1 \right) \left( 163138252419072t^{14} + 457946147463168t^{13} + 588427655607296t^{12} + 467272386885632t^{11} + 243746541248768t^{10} + 81184230845440t^{9} + 17211766560768t^{8} + 2390896372928t^{7} + 272364756944t^{6} + 38137945056t^{5} + 2404674180t^{4} + 1438456t^{3} + 13409741t^{2} + 33771t + 658 \right)D^{5} + \left( 1359485436825600t^{15} + 4714609200967680t^{14} + 7293185026566144t^{13} + 6816425183232000t^{12} + 4231501422768640t^{11} + 1729376772924672t^{10} + 446617382913920t^{9} + 74928047933632t^{8} + 9600940896288t^{7} + 1182654388880t^{6} + 98720920928t^{5} - 65952300t^{4} + 47181691t^{3} - 13168357t^{2} - 54357t - 329 \right)D^{4} + t \left( 5709838834667520t^{14} + 18279748010496000t^{13} + 26378666929607680t^{12} + 23236151238075392t^{11} + 13496888077032704t^{10} + 4991192078158080t^{9} + 1119885406945280t^{8} + 169136773118656t^{7} + 22922084504144t^{6} + 2857164316400t^{5} + 185097491972t^{4} + 2755005940t^{3} + 51292595t^{2} + 725196t + 259 \right)D^{3} + 2 t ^{2} \left( 6308012426870784t^{13} + 18908645957031936t^{12} + 25829730133600768t^{11} + 21743227297879552t^{10} + 11951361785838976t^{9} + 4040707135752832t^{8} + 805079083539328t^{7} + 116749005768256t^{6} + 16892507283448t^{5} + 1894154354776t^{4} + 88199781910t^{3} + 1072079356t^{2} + 31246885t + 269686 \right)D^{2} + 48 t ^{2} \left( 285491941733376t^{13} + 813123059284480t^{12} + 1065643756106112t^{11} + 867157291870208t^{10} + 455679304563584t^{9} + 142554426777056t^{8} + 25843259544832t^{7} + 3772080222664t^{6} + 566152350430t^{5} + 54866460788t^{4} + 1887825182t^{3} + 24965522t^{2} + 732804t + 4277 \right)D + 48 t ^{2} \left( 116527323156480t^{13} + 320124357580800t^{12} + 407541213829632t^{11} + 323839711229184t^{10} + 164485603770432t^{9} + 48362458139136t^{8} + 8174348162112t^{7} + 1227889352976t^{6} + 185762158528t^{5} + 15567985708t^{4} + 413750185t^{3} + 6948334t^{2} + 167062t + 658 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 7)

|λ| = 12.2283 λ ≈ 12.2283; multiplicity 1; Jordan blocks 1 λ ≈ -5.678364 - 4.683554i; multiplicity 1; Jordan blocks 1 λ ≈ -5.678364 + 4.683554i; multiplicity 1; Jordan blocks 1 λ ≈ -1.435784 - 2.26611i; multiplicity 1; Jordan blocks 1 λ ≈ -1.435784 + 2.26611i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
8
Spectral radius
$12.2282976$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.2283$11
$-5.678364 - 4.683554\mathrm{i}$11
$-5.678364 + 4.683554\mathrm{i}$11
$-1.435784 - 2.26611\mathrm{i}$11
$-1.435784 + 2.26611\mathrm{i}$11
$-2$11
$-1$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{5} + 2 \lambda^{4} - 80 \lambda^{3} - 912 \lambda^{2} - 2512 \lambda - 4768\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda^{5} + 2 \lambda^{4} - 80 \lambda^{3} - 912 \lambda^{2} - 2512 \lambda - 4768\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 18 & 18 & 40 & 48 & 80 \\ 1 & -1 & 3 & 0 & 0 & 18 & 18 & 16 \\ 1 & 1 & -1 & 4 & 8 & 12 & 18 & 16 \\ 0 & 1 & 4 & -1 & 2 & 12 & 12 & 12 \\ 0 & 2 & 1 & 2 & -1 & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & 5 & 2 & 6 & 0 \\ 0 & 0 & 0 & 1 & -2 & -1 & -4 & 4 \\ 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 3 & 0\end{pmatrix}\notag\end{equation}$$