Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-352-75-d

·

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

  • $\pi_1\colon X\to\mathbb P^2\times\mathrm{Gr}(2,4)$ is an isomorphism onto its image.

  • $\pi_2\colon X\to\mathbb P^2\times\mathrm{Gr}(2,4)$ is an isomorphism onto its image.

  • $\pi_3\colon X\to\mathbb P^2$ is the quadric surface fibration given by the linear projection of a smooth quadric fourfold (1-512-105-a) from a general plane; its discriminant is a smooth conic, over which the fibers are quadric cones.

  • $\pi_{12}\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a smooth conic, its intersection with a general plane.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

Defining data
Ambient
ℙ2P2 × ℙ2P2 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6 (marked)node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w4 + w6) + E^(w1 + w4)
Bundle
Q_2 box O_3(1) + O_1(1) box Q_2
ZL62 label
202031.wGw2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 75
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 352
$\chi(X, \mathrm{T}_X)$ = 6

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 352
$\mathrm{c}_1^2\mathrm{c}_2$ = 184
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 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 - 6 0 x0 - 6 x1
$\mathrm{HH}^{ 3 }$ = x1 + 60 0 0 x1 - 42 102
$\mathrm{HH}^{ 4 }$ = 75 0 0 0 0 75
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 10t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(137)
Coates et al. (2020) MW(10)
Coates–Kasprzyk–Prince (2015) CKP(186)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 4t^{2} + 1 \right) \left( 16384t^{6} + 512t^{4} + 44t^{2} - 1 \right) \left( 1473757184t^{8} + 13746176t^{6} + 7592448t^{4} + 46808t^{2} + 55 \right)D^{6} + 4 \left( 482920754053120t^{16} + 86447658893312t^{14} + 4886210281472t^{12} + 534951231488t^{10} + 18501230592t^{8} + 119944000t^{6} + 16234576t^{4} + 72412t^{2} + 55 \right)D^{5} + 4 \left( 3718489806209024t^{16} + 396693178679296t^{14} + 31496439267328t^{12} + 3577124749312t^{10} + 37524877312t^{8} - 202496928t^{6} - 28001956t^{4} - 105906t^{2} - 55 \right)D^{4} + 16 t ^{2} \left( 3525321504587776t^{14} + 219939516448768t^{12} + 31919813689344t^{10} + 2870602727424t^{8} + 30098031616t^{6} + 29514768t^{4} + 38264t^{2} + 755 \right)D^{3} + 64 t ^{2} \left( 1740023841947648t^{14} + 63218454626304t^{12} + 17856939016192t^{10} + 1163732019200t^{8} + 10148827776t^{6} + 77992960t^{4} + 145739t^{2} + 55 \right)D^{2} + 256 t ^{4} \left( 424064787152896t^{12} + 8915881033728t^{10} + 4878831042560t^{8} + 233990370304t^{6} + 2025100864t^{4} + 24310608t^{2} + 53911 \right)D + 30720 t ^{4} \left( 1320486436864t^{12} + 15835594752t^{10} + 16557547520t^{8} + 617585152t^{6} + 5801632t^{4} + 78192t^{2} + 187 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × Gr(2, 4)

|λ| = 7.600756 λ ≈ -7.600756; multiplicity 1; Jordan blocks 1 λ ≈ 7.600756; multiplicity 1; Jordan blocks 1 λ ≈ -2.230995 - 3.444284i; multiplicity 1; Jordan blocks 1 λ ≈ -2.230995 + 3.444284i; multiplicity 1; Jordan blocks 1 λ ≈ 2.230995 - 3.444284i; multiplicity 1; Jordan blocks 1 λ ≈ 2.230995 + 3.444284i; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
10
Spectral radius
$7.6007564$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-7.600756$11
$7.600756$11
$-2.230995 - 3.444284\mathrm{i}$11
$-2.230995 + 3.444284\mathrm{i}$11
$2.230995 - 3.444284\mathrm{i}$11
$2.230995 + 3.444284\mathrm{i}$11
$0 - 2\mathrm{i}$11
$0 + 2\mathrm{i}$11
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda^{2} + 4\right) \left(\lambda^{6} - 44 \lambda^{4} - 512 \lambda^{2} - 16384\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{2} + 4\right) \left(\lambda^{6} - 44 \lambda^{4} - 512 \lambda^{2} - 16384\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 0 & 0 & 0 & 0 & 8 & 4 & 0 \\ 2 & 0 & 0 & -2 & 4 & 2 & 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 4 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0 \\ 0 & 2 & 2 & 0 & 0 & 0 & 0 & -2 & 2 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 4 & 4 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0\end{pmatrix}\notag\end{equation}$$