Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-93-26-a

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
Fl(1, 2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w6) + E^(w1 + w2) + E^(3w2)
Bundle
wedge^4 Q_1^v ot U_2^v ot U_3^v + O(1,1) + O(0,3)
ZL62 label
602.5Dg
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,6) \\ \det\mathcal{E} &\cong \mathcal{O}(1,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 26
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 93
$\chi(X, \mathrm{T}_X)$ = -39

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 93
$\mathrm{c}_1^2\mathrm{c}_2$ = 114
$\mathrm{c}_2^2$ = 138
$\mathrm{c}_1\mathrm{c}_3$ = -6
$\mathrm{c}_4$ = 51
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 41 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 + x2 + 39 0 x0 + 39 x2
$\mathrm{HH}^{ 3 }$ = x1 + 16 0 0 x1 16
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 28 0 0 x1 - x2 + 2 0 26
$\mathrm{HH}^{ 5 }$ = 1 0 0 1 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 21t + 49t^{2} + 21t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(631)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(641)

Quantum differential equation

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

$$\begin{equation} L_X=\left( t + 1 \right) ^{3} \left( 12393t^{3} + 486t^{2} - 27t - 1 \right) \left( 27179113086t^{6} + 4122184446t^{5} + 124202646t^{4} - 28046331t^{3} + 12554373t^{2} + 21725t + 279 \right)D^{6} + 6 \left( t + 1 \right) ^{2} \left( 729799955028729t^{10} + 363568613381208t^{9} + 49147407702585t^{8} + 1430324946027t^{7} + 114781757130t^{6} + 221535275157t^{5} + 4015298709t^{4} - 19911190t^{3} + 8371878t^{2} + 8026t + 93 \right)D^{5} + \left( t + 1 \right) \left( 22567660147811466t^{11} + 22742874026213850t^{10} + 5657994642242214t^{9} + 500493326806413t^{8} - 8926651575081t^{7} + 11076370951023t^{6} + 3899852948268t^{5} + 37168582014t^{4} + 1429091731t^{3} - 45634005t^{2} - 80830t - 279 \right)D^{4} + 2 t \left( 29472690491544825t^{11} + 49124730458608218t^{10} + 24162282512778294t^{9} + 3854671232341491t^{8} + 160630520652441t^{7} + 5050318071420t^{6} + 19382082140295t^{5} + 2858803702977t^{4} + 16838507202t^{3} + 205277708t^{2} + 1298639t + 90 \right)D^{3} + 4 t ^{2} \left( 20546675656962678t^{10} + 29433342104070372t^{9} + 11762995374540342t^{8} + 1465302820178691t^{7} + 3654167086791t^{6} + 8427617296866t^{5} + 12214455213375t^{4} + 1069812647058t^{3} + 3809782561t^{2} + 105737489t + 466829 \right)D^{2} + 8 t ^{2} \left( 7241861092208157t^{10} + 9214270028899470t^{9} + 3094382366881704t^{8} + 318765343177773t^{7} - 10307627893998t^{6} + 4756292143128t^{5} + 3864516343272t^{4} + 200062392012t^{3} + 593648973t^{2} + 26650525t + 87420 \right)D + 144 t ^{2} \left( 112276916158266t^{10} + 130444042864680t^{9} + 37896489264960t^{8} + 3345911241066t^{7} - 211307501628t^{6} + 95478994944t^{5} + 54705901404t^{4} + 1635307299t^{3} + 6795702t^{2} + 296879t + 744 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 7)

|λ| = 29.04013 λ ≈ -29.04013; multiplicity 1; Jordan blocks 1 λ ≈ 21.70327; multiplicity 1; Jordan blocks 1 λ ≈ -19.66313; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 λ ≈ 0; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$29.0401344$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-29.04013$11
$21.70327$11
$-19.66313$11
$-1$31, 2
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{3} \left(\lambda^{3} + 27 \lambda^{2} - 486 \lambda - 12393\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{3} + 27 \lambda^{2} - 486 \lambda - 12393\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 24 & 36 & 18 & 1152 & 792 & 720 \\ 1 & -1 & 6 & 24 & -96 & 108 & 918 & 432 \\ 1 & 2 & -6 & 60 & 168 & -18 & -819 & 720 \\ 0 & 2 & 1 & -1 & 15 & 60 & -78 & 0 \\ 0 & 0 & 1 & 2 & -15 & 24 & 132 & 36 \\ 0 & 0 & 0 & 2 & 1 & -1 & 6 & 24 \\ 0 & 0 & 0 & 0 & 1 & 2 & -6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$