Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-114-30-c

·

Blow-up of a smooth quadric fourfold along a smooth minimal elliptic surface of degree $10$.

  • $\pi_1\colon X\to Y$ is the blow-up of a complete intersection $Y$ of three quadrics in $\mathbb P^7$ (a special member of 1-128-33-a) singular exactly along a line, along that line; the fibers over the line are quadric surfaces.

  • $\pi_2\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a smooth minimal elliptic surface of degree $10$ with $p_g=2$ and $q=0$, the rank-one locus of a general $2\times3$ matrix of forms of degrees $1,1,2$.

Varieties in this family can be blown down to

  • 1-512-105-a
  • A complete intersection of three quadrics in $\mathbb P^7$ singular along a line
Defining data
Ambient
ℙ5P5 × ℙ7P7
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_6 (marked)node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11node 12: fundamental weight ω_12ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12
Bundle
2 copies of E^(w1 + w6) + E^(w5 + w6) + E^(2w1)
Bundle
2 copies of O_1(1) box O_2(1) + Q_1 box O_2(1) + O_1(2)
ZL62 label
50700.w0cw04w04C
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6;8) \\ \det\mathcal{E} &\cong \mathcal{O}(5;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 114
$\chi(X, \mathrm{T}_X)$ = -29

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 114
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 42
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 32 2 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 + 29 0 x0 + 29 x1
$\mathrm{HH}^{ 3 }$ = x1 + 21 0 0 x1 + 4 17
$\mathrm{HH}^{ 4 }$ = 33 0 0 3 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 62t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(571)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(599)

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) ^{4} \left( 44t^{2} + 16t - 1 \right) \left( 148t^{2} + 24t + 1 \right) \left( 52087801392t^{8} + 119473528064t^{7} + 110350661120t^{6} + 52125776452t^{5} + 14304793677t^{4} + 1953947985t^{3} + 21641400t^{2} - 5724544t + 58368 \right)D^{6} + \left( t + 1 \right) ^{3} \left( 5766327965299968t^{13} + 18018720094434048t^{12} + 24358229220321920t^{11} + 18593538950262272t^{10} + 8931039083394752t^{9} + 2809007351430720t^{8} + 549817354104792t^{7} + 55292900431120t^{6} + 1406434050423t^{5} - 37367203899t^{4} + 7959896222t^{3} + 229648280t^{2} - 23890432t + 175104 \right)D^{5} + \left( t + 1 \right) ^{2} \left( 39007512706440960t^{14} + 140351663707832896t^{13} + 220541403664676160t^{12} + 198220418206289696t^{11} + 113335525514166384t^{10} + 43262349343379836t^{9} + 10873721447391392t^{8} + 1610399426690266t^{7} + 96319552824387t^{6} - 3570838015431t^{5} - 606481135637t^{4} - 18863281489t^{3} - 262234656t^{2} + 18116480t - 116736 \right)D^{4} + t \left( t + 1 \right) \left( 133982326252558080t^{14} + 563633959665110400t^{13} + 1040763691621468672t^{12} + 1109255322554640416t^{11} + 758743734979530720t^{10} + 350761683354250592t^{9} + 110675325755223336t^{8} + 22615443356110284t^{7} + 2532672796400223t^{6} + 75906932123609t^{5} - 8521314853925t^{4} - 553875671715t^{3} + 14327192896t^{2} + 206757760t - 198656 \right)D^{3} + 2 t \left( 122788866084622848t^{15} + 604433857053740384t^{14} + 1313208117113480544t^{13} + 1660204365662118144t^{12} + 1358921970657950504t^{11} + 759445024466138078t^{10} + 296088732926576288t^{9} + 79127209299066371t^{8} + 13336432263821078t^{7} + 1120825662717741t^{6} + 4846658458036t^{5} - 4165404444492t^{4} - 120775200836t^{3} + 10596680396t^{2} + 45278976t - 29184 \right)D^{2} + 4 t ^{2} \left( 56645692365005568t^{14} + 267417170435983392t^{13} + 556578932220628416t^{12} + 672951701662585504t^{11} + 525958664739302416t^{10} + 280252216260487906t^{9} + 103744737838490344t^{8} + 25960685619429143t^{7} + 3953630528140405t^{6} + 269332313515743t^{5} - 3558033729504t^{4} - 881181918714t^{3} + 5023512513t^{2} + 3156261488t + 3914304 \right)D + 96 t ^{3} \left( 847989406661760t^{13} + 3885990886732944t^{12} + 7841202026482440t^{11} + 9175401970660864t^{10} + 6928708787711878t^{9} + 3561412504793185t^{8} + 1266449290299084t^{7} + 300342138175645t^{6} + 41885540899476t^{5} + 2328077709462t^{4} - 61604680384t^{3} - 5725332102t^{2} + 377024544t + 29548800 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ5P5 × ℙ7P7

|λ| = 18.3923 λ ≈ 18.3923; multiplicity 1; Jordan blocks 1 λ ≈ -12 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -12 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2.392305; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 4; Jordan blocks 2, 2 λ ≈ 0; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$18.3923048$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.3923$11
$-12 - 2\mathrm{i}$11
$-12 + 2\mathrm{i}$11
$-2.392305$11
$-1$42, 2
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{4} \left(\lambda^{2} + 24 \lambda + 148\right) \left(\lambda^{2} - 16 \lambda - 44\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda^{2} + 24 \lambda + 148\right) \left(\lambda^{2} - 16 \lambda - 44\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 60 & 30 & 72 & 480 & 440 & 960 \\ 1 & -2 & 3 & 20 & 28 & 16 & 120 & 84 & 160 \\ 1 & 2 & -1 & 28 & 12 & 32 & 24 & 30 & 320 \\ 0 & 1 & 1 & -3 & -2 & 0 & 32 & 32 & 24 \\ 0 & 1 & 0 & 3 & -2 & 4 & 0 & 4 & 48 \\ 0 & 0 & 1 & 2 & 4 & -1 & 16 & 8 & 0 \\ 0 & 0 & 0 & 1 & -\frac{5}{4} & 2 & -6 & -\frac{7}{4} & 16 \\ 0 & 0 & 0 & 1 & 3 & 0 & 8 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{7}{4} & 0\end{pmatrix}\notag\end{equation}$$