Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-126-32-c

·

Blow-up of a smooth quadric fourfold along a del Pezzo surface of degree $2$.

  • $\pi_1\colon X\to Y$, onto the smooth del Pezzo threefold $Y=\mathrm{Gr}(2,5)\cap\mathbb P^6$ of degree $5$, is a conic bundle; its discriminant $\Delta\in|\mathcal O_Y(2)|$ is a surface of degree $10$ with $8$ ordinary double points whose minimal resolution is a K3 surface, and the fibers over the nodes are double lines.

  • $\pi_2\colon X\to\mathbb Q^4$ is the blow-up $\mathrm{Bl}_S\mathbb Q^4$ of the smooth quadric fourfold $\mathbb Q^4=\mathrm{Gr}(2,4)$ (1-512-105-a) along a smooth del Pezzo surface $S$ of degree $2$ with hyperplane class $-2K_S$.

Varieties in this family can be blown down to

Defining data
Ambient
Gr(2, 4) × Gr(2, 5)
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_7ω1ω2ω3ω4ω5ω6ω7
Bundle
3 copies of E^(w5) + E^(w2 + w7)
Bundle
3 copies of O_2(1) + O_1(1) box Q_2
ZL62 label
3141.444wA
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;5) \\ \det\mathcal{E} &\cong \mathcal{O}(3;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 32
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 126
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 126
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 114
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 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 + 11 0 x0 + 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 28 0 0 x1 + 17 11
$\mathrm{HH}^{ 4 }$ = 32 0 0 0 0 32
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 27t + 70t^{2} + 27t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(549)
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( 3t + 1 \right) ^{2} \left( 2000t^{5} - 1200t^{4} + 728t^{3} + 192t^{2} + 5t - 1 \right) \left( 229846743040t^{9} + 199368586368t^{8} + 80120825984t^{7} + 18267324384t^{6} + 1860677760t^{5} + 100429704t^{4} + 32728776t^{3} + 2564730t^{2} - 22500t + 189 \right)D^{6} + 2 \left( 3t + 1 \right) \left( 28960689623040000t^{16} + 29948231940608000t^{15} + 15219484475105280t^{14} + 8391371644238848t^{13} + 4884273991967744t^{12} + 2181264280878592t^{11} + 662965698269952t^{10} + 122090212321152t^{9} + 13767811617792t^{8} + 1995941572176t^{7} + 393656406912t^{6} + 36916313100t^{5} + 1038193380t^{4} + 56012388t^{3} + 5387499t^{2} - 34884t + 189 \right)D^{5} + \left( 1448034481152000000t^{17} + 1902901589928960000t^{16} + 1174761984572446720t^{15} + 519924747733067776t^{14} + 203105212097636864t^{13} + 74015010938779648t^{12} + 22824096821163008t^{11} + 4812344196502784t^{10} + 601657458772544t^{9} + 70935589391712t^{8} + 15666725411416t^{7} + 2244732683304t^{6} + 125180406690t^{5} + 1326662724t^{4} - 186532551t^{3} - 10591686t^{2} + 26676t - 189 \right)D^{4} + t \left( 6081744820838400000t^{16} + 7737465724999680000t^{15} + 4665827451910144000t^{14} + 1852579986033704960t^{13} + 550824974513872896t^{12} + 145370867955905536t^{11} + 37239696409958400t^{10} + 6933852741938688t^{9} + 803869052398848t^{8} + 146605777602624t^{7} + 37017219650304t^{6} + 4395744200448t^{5} + 190674507696t^{4} + 22403916t^{3} + 5770572t^{2} + 1216818t + 369 \right)D^{3} + t ^{2} \left( 13437759985090560000t^{15} + 16665515039003136000t^{14} + 9872632497786019840t^{13} + 3693026602635138048t^{12} + 927306645567797248t^{11} + 184256635739929088t^{10} + 38606397126544384t^{9} + 6516748673567616t^{8} + 752759931771648t^{7} + 185653518491280t^{6} + 44365546343008t^{5} + 4374562632804t^{4} + 145890973464t^{3} - 776073576t^{2} + 54156840t + 809613 \right)D^{2} + 4 t ^{2} \left( 3649046892503040000t^{15} + 4439860495350528000t^{14} + 2596348603140628480t^{13} + 940073400193013248t^{12} + 214500637465636864t^{11} + 34808976871533568t^{10} + 6131360177307136t^{9} + 993403314141120t^{8} + 121822369843200t^{7} + 33352464742776t^{6} + 7093102192224t^{5} + 603543526446t^{4} + 15420514656t^{3} - 98571771t^{2} + 10005750t + 70308 \right)D + 48 t ^{2} \left( 124117241241600000t^{15} + 149046577201920000t^{14} + 86399832547164160t^{13} + 30730783326243328t^{12} + 6656658776848384t^{11} + 956280409759744t^{10} + 150700296829440t^{9} + 24634374873792t^{8} + 3154841874176t^{7} + 830340544728t^{6} + 156481042432t^{5} + 12012003690t^{4} + 231834408t^{3} - 124761t^{2} + 212982t + 882 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 5)

|λ| = 17.84992 λ ≈ 17.84992; multiplicity 1; Jordan blocks 1 λ ≈ -7.196467 - 2.4177i; multiplicity 1; Jordan blocks 1 λ ≈ -7.196467 + 2.4177i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 2 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.7715095 - 1.161397i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7715095 + 1.161397i; multiplicity 1; Jordan blocks 1 2 2
Sector dimension
9
Spectral radius
$17.8499159$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$17.84992$11
$-7.196467 - 2.4177\mathrm{i}$11
$-7.196467 + 2.4177\mathrm{i}$11
$-3$22
$-2$21, 1
$0.7715095 - 1.161397\mathrm{i}$11
$0.7715095 + 1.161397\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda^{5} - 5 \lambda^{4} - 192 \lambda^{3} - 728 \lambda^{2} + 1200 \lambda - 2000\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda^{5} - 5 \lambda^{4} - 192 \lambda^{3} - 728 \lambda^{2} + 1200 \lambda - 2000\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 12 & 30 & 84 & 30 & 180 & 92 & 220 \\ 1 & 0 & 3 & 6 & 12 & 6 & 33 & 15 & 40 \\ 1 & 2 & -1 & 8 & 24 & 8 & 39 & 21 & 48 \\ 0 & 1 & -\frac{5}{2} & -\frac{7}{2} & -2 & -\frac{3}{2} & -14 & -6 & -12 \\ 0 & 1 & \frac{7}{2} & \frac{5}{2} & 4 & \frac{5}{2} & 24 & 12 & 24 \\ 0 & 1 & -\frac{5}{2} & -\frac{3}{2} & -2 & -\frac{7}{2} & -14 & -6 & -12 \\ 0 & 0 & 0 & 1 & 7 & 1 & \frac{13}{2} & \frac{9}{2} & 18 \\ 0 & 0 & 0 & 1 & -5 & 1 & -\frac{19}{2} & -\frac{15}{2} & -22 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$