Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-126-32-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ2P2 × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w2 + w3 + w5) + E^(w1 + w3 + w5)
Bundle
O_2(1) box O_3(1) box O_4(1) + O_1(1) box O_3(1) box O_4(1)
ZL62 label
112020.x4B00x4500
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 32
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 126
$\chi(X, \mathrm{T}_X)$ = -12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 126
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 117
$\mathrm{c}_1\mathrm{c}_3$ = 42
$\mathrm{c}_4$ = 27
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 17 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 12 0 x0 + 12 x1
$\mathrm{HH}^{ 3 }$ = x1 + 18 0 0 x1 + 10 8
$\mathrm{HH}^{ 4 }$ = 32 0 0 0 0 32
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 27t + 70t^{2} + 27t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(551)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(586)

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( 3t + 1 \right) \left( 5t - 1 \right) \left( 4t + 1 \right) ^{2} \left( 76t^{2} + 14t - 1 \right) \left( 83t^{3} + 39t^{2} + 8t + 1 \right) \left( 1918141129728t^{11} + 3205897797888t^{10} + 2364458073792t^{9} + 985920030784t^{8} + 258390945424t^{7} + 47962778416t^{6} + 7293410394t^{5} + 886251152t^{4} + 63921151t^{3} + 1451189t^{2} - 23801t + 231 \right)D^{6} + \left( 4t + 1 \right) \left( 15245539150368522240t^{19} + 39134603638188232704t^{18} + 46002316681181611008t^{17} + 32506192671936099072t^{16} + 15314994783438613376t^{15} + 5078551085198590592t^{14} + 1229894678389594240t^{13} + 224051382568160600t^{12} + 30758262907750236t^{11} + 2731969781420718t^{10} - 33031548628140t^{9} - 65896387171981t^{8} - 12649298358651t^{7} - 1334174942272t^{6} - 92024586939t^{5} - 5415204203t^{4} - 287839165t^{3} - 6169768t^{2} + 71403t - 462 \right)D^{5} + \left( 508184638345617408000t^{20} + 1373874087477778513920t^{19} + 1714203275309249421312t^{18} + 1299283469255635789824t^{17} + 666442134386643927808t^{16} + 245786657103965520512t^{15} + 68238104834821095552t^{14} + 14875649120339571104t^{13} + 2625625606439512984t^{12} + 371640888291922028t^{11} + 38587487351224418t^{10} + 2211283565627356t^{9} - 47885543271817t^{8} - 19130525935798t^{7} - 1134644818337t^{6} + 25021754407t^{5} + 7150360972t^{4} + 405210800t^{3} + 6795337t^{2} - 32301t + 231 \right)D^{4} + t \left( 2134375481051593113600t^{19} + 5579996929247498895360t^{18} + 6734632170820455751680t^{17} + 4934206025411255992320t^{16} + 2447591530579361977088t^{15} + 877031419823133237888t^{14} + 239387255723893768320t^{13} + 52253776870764884512t^{12} + 9385621439718069336t^{11} + 1360335462888862620t^{10} + 146648162977050618t^{9} + 9908422772222952t^{8} + 201269169847649t^{7} - 26492991238841t^{6} - 2270475808656t^{5} - 59486555603t^{4} - 310225531t^{3} - 21253347t^{2} - 1325908t - 517 \right)D^{3} + 2 t ^{2} \left( 2357976721923664773120t^{18} + 6003741001486509121536t^{17} + 7056140811160421265408t^{16} + 5028060161976622225920t^{15} + 2425006107473289857792t^{14} + 847619446406512286720t^{13} + 227712936074618850048t^{12} + 49526396395104643136t^{11} + 8923211671369889712t^{10} + 1292275818066014992t^{9} + 138875797289723384t^{8} + 9737186279253734t^{7} + 326579176526081t^{6} - 5204290603965t^{5} - 809596723953t^{4} - 22291641242t^{3} - 490686425t^{2} - 27667237t - 450471 \right)D^{2} + 8 t ^{2} \left( 640312644315477934080t^{18} + 1598321824960943075328t^{17} + 1840932499755607308288t^{16} + 1283786165672173936128t^{15} + 605542934917704403072t^{14} + 207505889327128345216t^{13} + 55039450810447442976t^{12} + 11921967235529877376t^{11} + 2143877745607110468t^{10} + 307550908629091494t^{9} + 32441912331384875t^{8} + 2246871259682699t^{7} + 82142588193014t^{6} + 206302872531t^{5} - 89158003781t^{4} - 2822640235t^{3} - 96708764t^{2} - 4726025t - 40656 \right)D + 16 t ^{2} \left( 130676049860301619200t^{18} + 321775459682215403520t^{17} + 365440347958576877568t^{16} + 250965790109889935616t^{15} + 116493039267910774656t^{14} + 39350008416348835776t^{13} + 10341669018803306784t^{12} + 2232724422082718384t^{11} + 400255672929406720t^{10} + 56813761684626240t^{9} + 5872716213060210t^{8} + 397000002357998t^{7} + 14523533951599t^{6} + 103654340223t^{5} - 10653645462t^{4} - 423832628t^{3} - 16451077t^{2} - 607915t - 3234 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ2P2 × ℙ2P2

|λ| = 18.18034 λ ≈ 18.18034; multiplicity 1; Jordan blocks 1 λ ≈ 5; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.209294 - 4.277176i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.209294 + 4.277176i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4.18034; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 6; Jordan blocks 1, 1, 1, 1, 1, 1 λ ≈ -3.581412; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 2 2 2 6 2 2
Sector dimension
18
Spectral radius
$18.1803399$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.18034$11
$5$21, 1
$-2.209294 - 4.277176\mathrm{i}$21, 1
$-2.209294 + 4.277176\mathrm{i}$21, 1
$-4.18034$11
$-4$61, 1, 1, 1, 1, 1
$-3.581412$21, 1
$-3$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda + 4\right)^{6} \left(\lambda - 5\right)^{2} \left(\lambda^{2} - 14 \lambda - 76\right) \left(\lambda^{3} + 8 \lambda^{2} + 39 \lambda + 83\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right) \left(\lambda - 5\right) \left(\lambda^{2} - 14 \lambda - 76\right) \left(\lambda^{3} + 8 \lambda^{2} + 39 \lambda + 83\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 10 & 10 & 6 & 21 & 21 & 21 & 21 & 48 & 27 & 27 & 48 & 48 & 104 & 104 & 220 \\ 1 & -1 & 0 & 1 & 1 & 4 & 8 & 8 & 2 & 2 & 8 & 6 & 6 & 18 & 18 & 36 & 24 & 72 \\ 1 & 0 & -1 & 1 & 1 & 4 & 2 & 2 & 8 & 8 & 8 & 6 & 6 & 18 & 18 & 24 & 36 & 72 \\ 1 & 1 & 1 & -1 & 2 & 0 & 2 & 4 & 2 & 4 & 8 & 2 & 6 & 3 & 6 & 15 & 15 & 32 \\ 1 & 1 & 1 & 2 & -1 & 0 & 4 & 2 & 4 & 2 & 8 & 6 & 2 & 6 & 3 & 15 & 15 & 32 \\ 0 & 1 & 1 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 6 & 6 & 8 & 8 & 24 \\ 0 & 1 & 0 & 1 & 0 & 1 & -2 & 2 & 0 & 0 & 1 & 0 & 1 & 2 & 4 & 6 & 2 & 12 \\ 0 & 1 & 0 & 0 & 1 & 1 & 2 & -2 & 0 & 0 & 1 & 1 & 0 & 4 & 2 & 6 & 2 & 12 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & -2 & 2 & 1 & 0 & 1 & 2 & 4 & 2 & 6 & 12 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 2 & -2 & 1 & 1 & 0 & 4 & 2 & 2 & 6 & 12 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 2 & -1 & 1 & 0 & 0 & 4 & 4 & 6 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 2 & 1 & -1 & 0 & 0 & 4 & 4 & 6 \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & -\frac{1}{2} & \frac{3}{2} & -\frac{1}{2} & -\frac{1}{3} & \frac{4}{3} & -\frac{5}{3} & -2 & 1 & \frac{2}{3} & \frac{2}{3} & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & -\frac{1}{2} & \frac{3}{2} & -\frac{1}{2} & \frac{3}{2} & -\frac{1}{3} & -\frac{5}{3} & \frac{4}{3} & 1 & -2 & \frac{2}{3} & \frac{2}{3} & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{3}{2} & 0 & 0 & 2 & 1 & 1 & \frac{3}{2} & \frac{3}{2} & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{3}{2} & 2 & 1 & 1 & \frac{3}{2} & \frac{3}{2} & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{3}{2} & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$