Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-90-25-c

·

This is a Fano fourfold of K3 type, appearing as case K3-35 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
Gr(2, 4) × Gr(2, 4)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w5) + E^(w2 + w4) + E^(w2 + w5)
Bundle
O_2(1) + O_1(1) box U_2^v + O_1(1) box O_2(1)
ZL62 label
3131.4nm
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 25
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 90
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 90
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 130
$\mathrm{c}_1\mathrm{c}_3$ = 20
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 24 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 + 23 0 x0 + 23 x2
$\mathrm{HH}^{ 3 }$ = x1 + 5 0 0 x1 5
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 13 0 0 x1 - x2 - 12 0 25
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 20t + 48t^{2} + 20t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(626)
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( 2t + 1 \right) \left( 4t + 1 \right) \left( 3t + 1 \right) ^{3} \left( 1408t^{4} - 1600t^{3} - 336t^{2} - 8t + 1 \right) \left( 2639213649920t^{9} + 4884353128960t^{8} + 3297889054080t^{7} + 1076647904160t^{6} + 177330936648t^{5} + 11826909148t^{4} - 376111436t^{3} - 88579860t^{2} - 1843095t + 18900 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 1872870460820029440t^{16} + 3402103823149301760t^{15} + 918777677935411200t^{14} - 2225208770047344640t^{13} - 2527411868716441600t^{12} - 1282939481955870720t^{11} - 378608836154155008t^{10} - 67724022082885632t^{9} - 6778542626728128t^{8} - 211537065373680t^{7} + 27997162307720t^{6} + 3074318618536t^{5} + 91350940536t^{4} + 2963655786t^{3} + 378586875t^{2} + 7674780t - 56700 \right)D^{5} + \left( 3t + 1 \right) \left( 46821761520500736000t^{17} + 101366438051158425600t^{16} + 64137179222288957440t^{15} - 13752446711085670400t^{14} - 41389638723600969728t^{13} - 26331387280121805824t^{12} - 9080734070730133248t^{11} - 1893384576089644064t^{10} - 228371886219331824t^{9} - 10271080326084352t^{8} + 1192270464621384t^{7} + 223627610283660t^{6} + 14742197451456t^{5} + 414972389132t^{4} + 454131463t^{3} - 456071580t^{2} - 5753805t + 37800 \right)D^{4} + t \left( 589954195158309273600t^{17} + 1480863745290063052800t^{16} + 1335567792967660339200t^{15} + 375527757397367439360t^{14} - 231577977755644354560t^{13} - 258199379301745921024t^{12} - 113964956170960673536t^{11} - 29295531523194820096t^{10} - 4541398196251902832t^{9} - 358920290293502256t^{8} + 4388706335610592t^{7} + 3977030869440992t^{6} + 366455694827316t^{5} + 12274247641002t^{4} - 36336823581t^{3} - 8802407142t^{2} - 117471045t + 193200 \right)D^{3} + 2 t \left( 651758920365370245120t^{17} + 1641914484673008107520t^{16} + 1544383924183423549440t^{15} + 593661915570992476160t^{14} - 46164401009713027072t^{13} - 145878499861257630720t^{12} - 68100810342713587968t^{11} - 16875569767734161136t^{10} - 2383622521740270976t^{9} - 147365451113211984t^{8} + 8404599845903656t^{7} + 2272167064374288t^{6} + 154493454889360t^{5} + 2597014004015t^{4} - 122947556378t^{3} - 4842360025t^{2} - 23669505t + 28350 \right)D^{2} + 12 t ^{2} \left( 117990839031661854720t^{16} + 298027544217692405760t^{15} + 287158230938927923200t^{14} + 126249358730671124480t^{13} + 12724731451247728640t^{12} - 13082410853577772032t^{11} - 7120451177401169152t^{10} - 1758374747768549872t^{9} - 232523955788824912t^{8} - 11375291663571288t^{7} + 1188852683719016t^{6} + 215954935527720t^{5} + 10631665464080t^{4} - 54511416941t^{3} - 17801700029t^{2} - 477431485t - 612720 \right)D + 288 t ^{3} \left( 2006646922307174400t^{15} + 5077508260066099200t^{14} + 4959647831145922560t^{13} + 2329465811056547840t^{12} + 415181501794695168t^{11} - 101113596930875648t^{10} - 75618805726323776t^{9} - 19180600376957608t^{8} - 2437703818311192t^{7} - 98349209318712t^{6} + 14230174189416t^{5} + 2010380055893t^{4} + 60102154744t^{3} - 3317736240t^{2} - 225211860t - 4847850 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 4)

|λ| = 24.37802 λ ≈ 24.37802; multiplicity 1; Jordan blocks 1 λ ≈ -8.567772 - 1.684505i; multiplicity 1; Jordan blocks 1 λ ≈ -8.567772 + 1.684505i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 3; Jordan blocks 1, 2 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.757525; multiplicity 1; Jordan blocks 1 3 2
Sector dimension
10
Spectral radius
$24.3780194$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$24.37802$11
$-8.567772 - 1.684505\mathrm{i}$11
$-8.567772 + 1.684505\mathrm{i}$11
$-4$11
$-3$31, 2
$-2$21, 1
$0.757525$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{3} \left(\lambda + 4\right) \left(\lambda^{4} - 8 \lambda^{3} - 336 \lambda^{2} - 1600 \lambda + 1408\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda + 4\right) \left(\lambda^{4} - 8 \lambda^{3} - 336 \lambda^{2} - 1600 \lambda + 1408\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 20 & 87 & 192 & 87 & 54 & 660 & 380 & 1120 \\ 1 & 0 & 3 & 16 & 36 & 16 & 8 & 96 & 57 & 184 \\ 1 & 4 & -1 & 12 & 32 & 12 & 12 & 117 & 69 & 192 \\ 0 & 1 & 0 & -2 & 3 & 0 & 1 & 8 & 6 & 12 \\ 0 & 1 & 1 & 2 & -1 & 2 & 1 & 20 & 10 & 30 \\ 0 & 1 & 0 & 0 & 3 & -2 & 1 & 8 & 6 & 12 \\ 0 & 0 & 2 & 3 & 8 & 3 & -2 & 14 & 8 & 36 \\ 0 & 0 & 0 & 1 & 6 & 1 & \frac{16}{5} & \frac{43}{5} & \frac{36}{5} & \frac{104}{5} \\ 0 & 0 & 0 & 1 & -4 & 1 & -\frac{18}{5} & -\frac{49}{5} & -\frac{48}{5} & -\frac{72}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$