Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-102-27-a

·

Küchle’s fourfold of type (c7), the blow-up of a cubic fourfold along a Veronese surface.

  • $f\colon X\to Y$ is the blow-up of a general smooth cubic fourfold $Y\subset\mathbb P^5$ containing a Veronese surface (1-243-55-a), along that surface.

  • $f'\colon X\to Y'$ is the blow-up of another smooth cubic fourfold $Y'\subset\mathbb P^5$ (1-243-55-a) along a Veronese surface, obtained from the first through the quadratic Cremona transformation associated with the Veronese surface.

Varieties in this family can be blown down to

Defining data
Ambient
Gr(3, 8)
node 1: fundamental weight ω_1node 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_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w3) + E^(w5)
Bundle
O(1) + wedge^2 Q^v(1)
ZL62 label
703.42
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(8) \\ \det\mathcal{E} &\cong \mathcal{O}(7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 102
$\chi(X, \mathrm{T}_X)$ = -19

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 102
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 132
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 30
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 22 1 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 + x2 + 19 0 x0 + 19 x2
$\mathrm{HH}^{ 3 }$ = x1 + 9 0 0 x1 9
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 19 0 0 x1 - x2 - 8 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 56t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 24t - 17 \right) ^{2} \left( t^{2} - 11t - 1 \right) \left( 64t^{2} + 16t - 1 \right)D^{4} + 2 t \left( 24t - 17 \right) \left( 4608t^{4} - 37376t^{3} + 29304t^{2} + 8194t + 61 \right)D^{3} + t \left( 479232t^{5} - 3246336t^{4} + 4925056t^{3} - 2002124t^{2} - 423285t - 1326 \right)D^{2} + t \left( 442368t^{5} - 2434560t^{4} + 3712032t^{3} - 2131188t^{2} - 287674t - 289 \right)D + 8 t ^{2} \left( 18432t^{4} - 86304t^{3} + 125748t^{2} - 93381t - 9248 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(3, 8)

|λ| = 19.31371 λ ≈ 19.31371; multiplicity 1; Jordan blocks 1 λ ≈ -11.09017; multiplicity 1; Jordan blocks 1 λ ≈ -3.313708; multiplicity 1; Jordan blocks 1 λ ≈ 0.09016994; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 2 2
Sector dimension
6
Spectral radius
$19.3137085$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.31371$11
$-11.09017$11
$-3.313708$11
$0.09016994$11
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda^{2} + 11 \lambda - 1\right) \left(\lambda^{2} - 16 \lambda - 64\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda^{2} + 11 \lambda - 1\right) \left(\lambda^{2} - 16 \lambda - 64\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 32 & 108 & 168 & 240 & 960 \\ 1 & 1 & 36 & 56 & 45 & 240 \\ 0 & 1 & -3 & -6 & -4 & -12 \\ 0 & 1 & 3 & 6 & 12 & 36 \\ 0 & 0 & 7 & 10 & 1 & 32 \\ 0 & 0 & 0 & 0 & 1 & 0\end{pmatrix}\notag\end{equation}$$