Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-136-34-a

·

Blow-up of a codimension-two linear section of $\operatorname{LG}(3,6)$ along a del Pezzo surface of degree $4$.

  • $\pi_1\colon X\to\mathrm{Gr}(3,6)$ is the blow-up of a smooth codimension-two linear section $W$ of $\mathrm{LG}(3,6)$, a Fano fourfold of index $2$ and degree $16$ (1-256-57-a), along a smooth quartic del Pezzo surface.

  • $\pi_2\colon X\to\mathbb P^5$ maps onto a linear $\mathbb P^3$ as a conic bundle whose discriminant is a quartic surface with eight ordinary double points.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 3, 6)
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
2 copies of E^(w1) + 2 copies of E^(w3) + E^(w4)
Bundle
2 copies of O(1,0) + 2 copies of O(0,1) + Q_1^v ot wedge^2 U_2^v ot U_3^v
ZL62 label
54.32200
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,5) \\ \det\mathcal{E} &\cong \mathcal{O}(2,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 34
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 136
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 136
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 112
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 0 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 + x1 + 11 0 x0 + 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 30 0 0 x1 + 15 15
$\mathrm{HH}^{ 4 }$ = 34 0 0 0 0 34
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 29t + 76t^{2} + 29t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
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( 3t + 1 \right) ^{3} \left( 72t^{2} + 12t - 1 \right) \left( 8t^{3} + 52t^{2} + 7t + 1 \right) \left( 19036020287232t^{11} + 43281623972160t^{10} + 41681103724368t^{9} + 22491261109112t^{8} + 7578263204324t^{7} + 1681397217776t^{6} + 252164679422t^{5} + 25798259844t^{4} + 1778781184t^{3} + 71683195t^{2} + 407077t + 4114 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 1381558208366149632t^{18} + 11547243440991830016t^{17} + 28821410283346126848t^{16} + 37045358085770210304t^{15} + 29444015566957105152t^{14} + 15816303987608590848t^{13} + 6040384012784172192t^{12} + 1692335719842128208t^{11} + 355036326618504440t^{10} + 56621380928002156t^{9} + 6964820276209924t^{8} + 671963763472546t^{7} + 51697122038196t^{6} + 3223391705558t^{5} + 169542198736t^{4} + 7953088651t^{3} + 282663682t^{2} + 1147179t + 8228 \right)D^{5} + \left( 3t + 1 \right) \left( 34538955209153740800t^{19} + 261081065831358136320t^{18} + 654193018067610875904t^{17} + 868219844249602874880t^{16} + 723878881302569408640t^{15} + 413022298569250492800t^{14} + 169496030744982648384t^{13} + 51614804662302466432t^{12} + 11898866409969442608t^{11} + 2101568450801788668t^{10} + 285461660766981976t^{9} + 29530601669941426t^{8} + 2234288814773350t^{7} + 109677101612230t^{6} + 2151199524947t^{5} - 91396010850t^{4} - 7359077720t^{3} - 262781852t^{2} - 1055549t - 4114 \right)D^{4} + t \left( 435190835635337134080t^{19} + 3047466289248318504960t^{18} + 7732572440893158021120t^{17} + 10693242990357452789760t^{16} + 9442332699980884178688t^{15} + 5777572009421623795200t^{14} + 2571312143959958429280t^{13} + 858508139709609911280t^{12} + 219464342471663262408t^{11} + 43521964799226219092t^{10} + 6742204400058523116t^{9} + 815386086599086810t^{8} + 75903216297146324t^{7} + 5227269033594120t^{6} + 245260005967218t^{5} + 6820972685639t^{4} + 100327376838t^{3} + 1330024990t^{2} + 13938276t + 6897 \right)D^{3} + 2 t ^{2} \left( 480782256511420071936t^{18} + 3039432621937761153024t^{17} + 7266182168260622075904t^{16} + 9572036158444534306560t^{15} + 8081440373936608220736t^{14} + 4735084594949528360256t^{13} + 2019933425956499715024t^{12} + 647123746345033876512t^{11} + 158935094768207508632t^{10} + 30318115344312892712t^{9} + 4519014282345826540t^{8} + 524381407797751456t^{7} + 46382696685715956t^{6} + 2968180103367067t^{5} + 124351369565864t^{4} + 2958123656433t^{3} + 43949555526t^{2} + 790516934t + 5433472 \right)D^{2} + 16 t ^{2} \left( 65278625345300570112t^{18} + 380116214275410330624t^{17} + 865196794813635624960t^{16} + 1095817494844177807680t^{15} + 892697375483569551024t^{14} + 505507053865914889344t^{13} + 208640052739725970680t^{12} + 64743537193639791516t^{11} + 15420774733518107418t^{10} + 2855543176383357223t^{9} + 413087244455708686t^{8} + 46364208402751360t^{7} + 3926354420573569t^{6} + 235258187813959t^{5} + 8865663151603t^{4} + 181462698844t^{3} + 2749393195t^{2} + 50241829t + 267410 \right)D + 16 t ^{2} \left( 26644336875632885760t^{18} + 146169448495871569920t^{17} + 320618628925763309568t^{16} + 394221812089410222336t^{15} + 312654108868556738112t^{14} + 172588354189526823264t^{13} + 69501640304337732792t^{12} + 21062159322006686544t^{11} + 4903852907889278544t^{10} + 888247965302612904t^{9} + 125631862140282716t^{8} + 13740876864495529t^{7} + 1123612890231836t^{6} + 63766193519512t^{5} + 2199908895688t^{4} + 40062312588t^{3} + 637187046t^{2} + 9982137t + 41140 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 6)

|λ| = 16.3923 λ ≈ 16.3923; multiplicity 1; Jordan blocks 1 λ ≈ -3.421453 - 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -3.421453 + 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -4.392305; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 3; Jordan blocks 1, 2 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -0.1570937; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$16.3923048$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$16.3923$11
$-3.421453 - 6.262482\mathrm{i}$11
$-3.421453 + 6.262482\mathrm{i}$11
$-4.392305$11
$-3$31, 2
$-2$11
$-0.1570937$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{3} \left(\lambda^{2} - 12 \lambda - 72\right) \left(\lambda^{3} + 7 \lambda^{2} + 52 \lambda + 8\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda^{2} - 12 \lambda - 72\right) \left(\lambda^{3} + 7 \lambda^{2} + 52 \lambda + 8\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 12 & 72 & 30 & 48 & 72 & 224 & 160 \\ 1 & -1 & 4 & 4 & 0 & 8 & 6 & 24 & 32 \\ 1 & 1 & 0 & 16 & 8 & 8 & 18 & 60 & 40 \\ 0 & 1 & 1 & -1 & 0 & 4 & 0 & 0 & 6 \\ 0 & 1 & 0 & 4 & -1 & 0 & 4 & 8 & 0 \\ 0 & 0 & 2 & 2 & 1 & -2 & 4 & 16 & 12 \\ 0 & 0 & 0 & -2 & -3 & 4 & -4 & -4 & 4 \\ 0 & 0 & 0 & \frac{7}{2} & 2 & 1 & 2 & 3 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 6 & 0\end{pmatrix}\notag\end{equation}$$