Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-48-17-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ6P6
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(3w3) + E^(w1 + w3) + E^(w2 + w3)
Bundle
O_2(3) + O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
20600.Iw7w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 17
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 48
$\chi(X, \mathrm{T}_X)$ = -57

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 48
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 171
$\mathrm{c}_1\mathrm{c}_3$ = -30
$\mathrm{c}_4$ = 99
Hodge diamond
1
00
0 2 0
0 0 0 0
0 8 77 8 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 + 57 0 x0 + 57 x2
$\mathrm{HH}^{ 3 }$ = x1 + 4 0 0 x1 4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 46 0 0 x1 - x2 + 29 0 17
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 12t + 22t^{2} + 12t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(712)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(709)

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( 7t + 1 \right) ^{2} \left( 6t + 1 \right) ^{4} \left( 18729t^{3} - 795t^{2} + 62t - 1 \right) \left( 19640828626820802t^{10} + 20965510520922207t^{9} + 9048192819550408t^{8} + 2038691002773275t^{7} + 257675705364226t^{6} + 18451253038979t^{5} + 778909616661t^{4} + 26073623204t^{3} + 736305645t^{2} - 1609427t + 144 \right)D^{6} + \left( 7t + 1 \right) \left( 6t + 1 \right) ^{3} \left( 324446415988223038180356t^{15} + 400964244088518361219284t^{14} + 208207113226091136369837t^{13} + 59527144524457433993184t^{12} + 10314721533368698388808t^{11} + 1130647302389303369725t^{10} + 81162133111439854283t^{9} + 4098369637865938125t^{8} + 162640476871065000t^{7} + 5142149513776905t^{6} + 140395901871822t^{5} + 4399580278628t^{4} + 62948259037t^{3} + 2998960359t^{2} - 4825977t + 288 \right)D^{5} + \left( 6t + 1 \right) ^{2} \left( 113556245595878063363124600t^{17} + 166521226882046307011504820t^{16} + 106213854882384383394976182t^{15} + 38887085972028170244663279t^{14} + 9073070137114427202758025t^{13} + 1417305119086911810011304t^{12} + 152232858790538662381919t^{11} + 11507606237369782018694t^{10} + 637369548197121736615t^{9} + 27435015233500744872t^{8} + 906573961183643823t^{7} + 17585395811291272t^{6} + 121448995141t^{5} - 6427558042026t^{4} - 147197790854t^{3} - 2270734707t^{2} + 3274105t - 144 \right)D^{4} + t \left( 6t + 1 \right) \left( 2861617389016127196750739920t^{17} + 4495771667366601021496338480t^{16} + 3103398576768669720904535400t^{15} + 1244494024699950349859987496t^{14} + 322635484931142052701244233t^{13} + 56957220189774683970888567t^{12} + 7043592137142220634988372t^{11} + 622912399497626743412505t^{10} + 40524593279851471588358t^{9} + 2031987096043873476276t^{8} + 81290303206602426401t^{7} + 2411299686813408831t^{6} + 40667098122556325t^{5} + 139233808114470t^{4} - 4465107801489t^{3} - 71799700230t^{2} + 56455380t + 1485 \right)D^{3} + 6 t ^{2} \left( 6322811754778490568058777728t^{17} + 10687257852953585483277226368t^{16} + 8014773110706185640539194368t^{15} + 3530498619705593612742103968t^{14} + 1018337579170641119366710560t^{13} + 202977685484533069586042616t^{12} + 28801080388767842791390144t^{11} + 2966519356394999218724687t^{10} + 226389708075069412222719t^{9} + 13211729613283229411860t^{8} + 611219300972636960993t^{7} + 22121844309628479464t^{6} + 543112948747729266t^{5} + 6039655343498535t^{4} - 27667039983895t^{3} - 908261148504t^{2} - 3567193523t + 2772256 \right)D^{2} + 12 t ^{2} \left( 3433940866819352636100887904t^{17} + 5684863939791879285775085040t^{16} + 4158283470615038250062144352t^{15} + 1779224770187713234733427144t^{14} + 496398261148408851675137514t^{13} + 95306392595449938944318853t^{12} + 12979396167718000161387861t^{11} + 1280951124224905282515237t^{10} + 93918520765133401034216t^{9} + 5309664158857412009633t^{8} + 239079794653040079326t^{7} + 8229780879648614313t^{6} + 177644959212001086t^{5} + 1218250551005722t^{4} - 21956525117367t^{3} - 304533040926t^{2} - 125554780t + 39744 \right)D + 48 t ^{2} \left( 350402129267280881234784480t^{17} + 571855342577851607021636400t^{16} + 411051649906786891287250656t^{15} + 172272872838822272928112896t^{14} + 46918413590029288587604404t^{13} + 8763256806292168843310256t^{12} + 1157473902881935659437250t^{11} + 110633356209108891012631t^{10} + 7874063567720248475353t^{9} + 434914820348357841922t^{8} + 19157125723748502025t^{7} + 630414329624076890t^{6} + 12119438260915902t^{5} + 42580326573515t^{4} - 1994045808575t^{3} - 20955839724t^{2} - 631575t + 3744 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ6P6

|λ| = 53.69 λ ≈ 53.69; multiplicity 1; Jordan blocks 1 λ ≈ 4.155002 - 18.20912i; multiplicity 1; Jordan blocks 1 λ ≈ 4.155002 + 18.20912i; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 2; Jordan blocks 2 λ ≈ -6; multiplicity 4; Jordan blocks 1, 3 2 4
Sector dimension
9
Spectral radius
$53.6899963$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$53.69$11
$4.155002 - 18.20912\mathrm{i}$11
$4.155002 + 18.20912\mathrm{i}$11
$-7$22
$-6$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{4} \left(\lambda + 7\right)^{2} \left(\lambda^{3} - 62 \lambda^{2} + 795 \lambda - 18729\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right)^{3} \left(\lambda + 7\right)^{2} \left(\lambda^{3} - 62 \lambda^{2} + 795 \lambda - 18729\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 132 & 1188 & 432 & 2376 & 32256 & 12672 & 241920 \\ 1 & -1 & 12 & 108 & 0 & 288 & 1620 & 432 & 13824 \\ 1 & 1 & 9 & 84 & 42 & 192 & 3078 & 1323 & 25344 \\ 0 & 1 & 1 & 8 & -1 & 30 & 108 & 0 & 648 \\ 0 & 1 & 0 & 12 & 0 & 12 & 288 & 108 & 1728 \\ 0 & 0 & 1 & 1 & 1 & 0 & 84 & 42 & 864 \\ 0 & 0 & 0 & 1 & 0 & 3 & 1 & 1 & 48 \\ 0 & 0 & 0 & 1 & 1 & -2 & 28 & 7 & 168 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$