Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-378-80-a

·

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

  • $\pi_1\colon X\to\mathrm{Gr}(2,4)$ is the blow-up of the quadric fourfold $\mathrm{Gr}(2,4)$ (1-512-105-a) along a smooth quadric surface, a codimension-two linear section of it.

  • $\pi_2\colon X\to\mathbb P^1$ is a fibration in quadric threefolds, the hyperplane sections of the quadric fourfold $\mathrm{Gr}(2,4)$ (1-512-105-a) through the quadric surface; exactly two fibers are quadric cones.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(2, 3, 8)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w2) + 2 copies of E^(w7) + E^(w1 + w3)
Bundle
O(1,0) + 2 copies of wedge^4 Q_1^v ot U_2^v ot wedge^2 U_3^v + U_2^v ot Sigma_(2,1) U_3^v
ZL62 label
705.661w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,6) \\ \det\mathcal{E} &\cong \mathcal{O}(2,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 80
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 378
$\chi(X, \mathrm{T}_X)$ = 7

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 378
$\mathrm{c}_1^2\mathrm{c}_2$ = 192
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 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 - 7 0 x0 - 7 x1
$\mathrm{HH}^{ 3 }$ = x1 + 64 0 0 x1 - 49 113
$\mathrm{HH}^{ 4 }$ = 80 0 0 0 0 80
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 75t + 226t^{2} + 75t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(19)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(26)

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( 7t - 1 \right) \left( 20t^{2} + 8t + 1 \right) \left( 76t^{3} - 36t^{2} - t - 1 \right) \left( 139643935200t^{9} + 340182037440t^{8} + 232696678780t^{7} - 5565754536t^{6} - 3792871221t^{5} - 2458348617t^{4} - 23073750t^{3} - 9171738t^{2} - 19521t - 1701 \right)D^{6} + \left( 25258794998976000t^{16} + 80284597780262400t^{15} + 81946633769053760t^{14} + 13724130000552000t^{13} - 14914523821328976t^{12} - 3820312677064080t^{11} - 264482127823524t^{10} + 180754040811420t^{9} + 39451766812875t^{8} + 2127097690191t^{7} + 289777355013t^{6} + 57423273891t^{5} + 14788175823t^{4} + 118733085t^{3} + 36597771t^{2} + 48357t + 3402 \right)D^{5} + \left( 170868319110720000t^{16} + 538045449955977600t^{15} + 548695777032258240t^{14} + 109390901993285360t^{13} - 58447198555881552t^{12} - 13662873754763100t^{11} - 1508197204723144t^{10} + 878900071282858t^{9} + 179522092645017t^{8} + 17713653928810t^{7} + 948101308074t^{6} + 25065209022t^{5} - 21925357428t^{4} - 67877910t^{3} - 35759826t^{2} - 37584t - 1701 \right)D^{4} + t \left( 586895530858560000t^{15} + 1836218696608800000t^{14} + 1854577999769467840t^{13} + 364543739269602240t^{12} - 124981608451577712t^{11} - 34716234074940240t^{10} - 6837892382117844t^{9} + 2072106482895336t^{8} + 300293993910843t^{7} + 22045863435597t^{6} + 2409503378127t^{5} + 96546561393t^{4} + 927207009t^{3} + 28946871t^{2} + 729t + 243 \right)D^{3} + 2 t ^{2} \left( 537863752331136000t^{14} + 1675961752799073600t^{13} + 1673430591138420800t^{12} + 297838863010840120t^{11} - 78320571066554928t^{10} - 31382562563784870t^{9} - 7765013269555446t^{8} + 1344874179110929t^{7} + 137038373895603t^{6} + 13288789371413t^{5} + 1597736569503t^{4} + 49433360067t^{3} + 660632529t^{2} + 20297925t - 189 \right)D^{2} + 36 t ^{3} \left( 27570057286464000t^{13} + 85707394671729600t^{12} + 84713638774222080t^{11} + 13114773201735240t^{10} - 3050367883701296t^{9} - 1780220208725210t^{8} - 452445225115394t^{7} + 49677554080863t^{6} + 3037289834646t^{5} + 542065674339t^{4} + 61702680042t^{3} + 1463792877t^{2} + 27341874t + 719523 \right)D + 144 t ^{3} \left( 2476352450880000t^{13} + 7689006515937600t^{12} + 7540832647443200t^{11} + 1006874833378520t^{10} - 229364267948400t^{9} - 178422255768990t^{8} - 43732986730902t^{7} + 3298348794321t^{6} + 69130445364t^{5} + 43820206119t^{4} + 4381375698t^{3} + 81639495t^{2} + 2076192t + 45927 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 4)

|λ| = 7 λ ≈ 7; multiplicity 1; Jordan blocks 1 λ ≈ -1.421453 - 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -1.421453 + 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -4 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -4 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ 1.842906; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
9
Spectral radius
$7$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7$11
$-1.421453 - 6.262482\mathrm{i}$11
$-1.421453 + 6.262482\mathrm{i}$11
$-4 - 2\mathrm{i}$11
$-4 + 2\mathrm{i}$11
$1.842906$11
$-1$11
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right) \left(\lambda - 7\right) \left(\lambda^{2} + 8 \lambda + 20\right) \left(\lambda^{3} + \lambda^{2} + 36 \lambda - 76\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda - 7\right) \left(\lambda^{2} + 8 \lambda + 20\right) \left(\lambda^{3} + \lambda^{2} + 36 \lambda - 76\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 3 & 3 & 4 & 4 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 3 & 3 & 0 \\ 3 & 1 & 0 & 0 & 0 & 0 & 0 & 3 & 4 \\ 0 & 3 & 1 & -1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & 1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 3 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$