Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-60-19-c

·

This is a Fano fourfold of K3 type, appearing as case K3-24 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × Gr(2, 5)
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_5ω1ω2ω3ω4ω5
Bundle
E^(w2 + w3) + E^(w1 + w3)
Bundle
U_2^v(1) + O_1(1) box O_2(1)
ZL62 label
141.da
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 19
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 60
$\chi(X, \mathrm{T}_X)$ = -31

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 60
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 144
$\mathrm{c}_1\mathrm{c}_3$ = 4
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 32 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 + 31 0 x0 + 31 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 9 0 0 x1 - x2 - 10 0 19
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 14t + 30t^{2} + 14t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(699)
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( 5t + 1 \right) \left( 6t + 1 \right) ^{3} \left( 92t^{2} + 12t - 1 \right) \left( 188t^{2} + 36t - 1 \right) \left( 244125376t^{6} + 164617632t^{5} + 34407264t^{4} + 2402168t^{3} - 11096t^{2} - 3064t + 5 \right)D^{6} + 2 \left( 6t + 1 \right) ^{2} \left( 1330053638538240t^{12} + 1628217532440576t^{11} + 803400495787264t^{10} + 204004222886144t^{9} + 27566576158080t^{8} + 1706419343936t^{7} + 4193784160t^{6} - 3752358640t^{5} - 80456492t^{4} - 1406820t^{3} + 24501t^{2} + 4486t - 5 \right)D^{5} + \left( 6t + 1 \right) \left( 133005363853824000t^{13} + 170227590003824640t^{12} + 89591834749257728t^{11} + 25146865352033792t^{10} + 4027037345131008t^{9} + 354678860039936t^{8} + 13272961497216t^{7} - 193385742688t^{6} - 24316582464t^{5} + 60529976t^{4} + 29853176t^{3} + 29132t^{2} - 5537t + 5 \right)D^{4} + 2 t \left( 1675867584558182400t^{13} + 2278037831704104960t^{12} + 1298025071740108800t^{11} + 405729488434501632t^{10} + 75658615778430208t^{9} + 8439526740161664t^{8} + 511274721119872t^{7} + 9540871186912t^{6} - 609899688880t^{5} - 33424526584t^{4} - 343447864t^{3} + 4015882t^{2} - 88089t - 18 \right)D^{3} + 4 t ^{2} \left( 1851434664845230080t^{12} + 2393165417441734656t^{11} + 1282970126513299968t^{10} + 373704535556573696t^{9} + 64174094065571840t^{8} + 6460613450407040t^{7} + 334753306021952t^{6} + 2927688203552t^{5} - 537130587536t^{4} - 22374396744t^{3} - 213840422t^{2} + 486328t - 25035 \right)D^{2} + 16 t ^{2} \left( 502760275367454720t^{12} + 625287061723742208t^{11} + 319457579962261248t^{10} + 87902760876845568t^{9} + 14096012148746048t^{8} + 1297459929132800t^{7} + 57624519502128t^{6} - 90890542704t^{5} - 116593048476t^{4} - 3955912344t^{3} - 31595254t^{2} - 95257t - 1875 \right)D + 32 t ^{2} \left( 102604137830092800t^{12} + 124220433133854720t^{11} + 61300175350381056t^{10} + 16170977630921472t^{9} + 2460315683076800t^{8} + 210559318700352t^{7} + 8103884424000t^{6} - 96932885344t^{5} - 19788557884t^{4} - 579419120t^{3} - 4013610t^{2} - 28222t - 145 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 5)

|λ| = 40.62742 λ ≈ 40.62742; multiplicity 1; Jordan blocks 1 λ ≈ 17.31371; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 3; Jordan blocks 1, 2 λ ≈ -5.313708; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4.627417; multiplicity 1; Jordan blocks 1 3 2
Sector dimension
9
Spectral radius
$40.627417$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$40.62742$11
$17.31371$11
$-6$31, 2
$-5.313708$11
$-5$21, 1
$-4.627417$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{2} \left(\lambda + 6\right)^{3} \left(\lambda^{2} - 12 \lambda - 92\right) \left(\lambda^{2} - 36 \lambda - 188\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda + 6\right)^{2} \left(\lambda^{2} - 12 \lambda - 92\right) \left(\lambda^{2} - 36 \lambda - 188\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 106 & 288 & 870 & 1218 & 2880 & 9120 & 28200 \\ 1 & -1 & 5 & 96 & 70 & 98 & 750 & 1200 & 6240 \\ 1 & 1 & 7 & 24 & 100 & 140 & 270 & 1020 & 2880 \\ 0 & 1 & 1 & 6 & 5 & 7 & 90 & 120 & 750 \\ 0 & 0 & 1 & 1 & 0 & 7 & 10 & 50 & 120 \\ 0 & 0 & 1 & 1 & 5 & 2 & 10 & 50 & 120 \\ 0 & 0 & 0 & \frac{12}{5} & 1 & \frac{7}{5} & 6 & 12 & 96 \\ 0 & 0 & 0 & 0 & 1 & \frac{7}{5} & 1 & 0 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$