Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-176-42-a

·

A conic bundle over a smooth quadric threefold.

  • $\pi_1\colon X\to Q^3$, onto a smooth quadric threefold in $\mathbb P^4$, is a conic bundle; its discriminant is a complete intersection of a quadric and a cubic in $\mathbb P^4$ with eight ordinary nodes, whose minimal resolution is a K3 surface, and the fibers over the nodes are double lines.

  • $\pi_2\colon X\to\mathbb P^2$ is a quadric surface fibration whose discriminant is a smooth plane quartic, over which the fibers are quadric cones.

Defining data
Ambient
ℙ2P2 × ℙ4P4
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(2w3) + E^(2w1 + w3)
Bundle
O_2(2) + O_1(2) box O_2(1)
ZL62 label
2040.8x317
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 42
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 176
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 176
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 38 0 0 x1 + 7 31
$\mathrm{HH}^{ 4 }$ = 42 0 0 0 0 42
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 37t + 100t^{2} + 37t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(354)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(430)

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( 2t + 1 \right) ^{2} \left( 6704t^{6} + 6048t^{5} + 4416t^{4} + 1080t^{3} + 60t^{2} - 1 \right) \left( 56344745156336t^{13} + 149098961230416t^{12} - 31508013300344t^{11} - 293509357229988t^{10} - 311027871386192t^{9} - 157442177947632t^{8} - 41167025199224t^{7} - 4904500264644t^{6} - 147540125306t^{5} + 5428150131t^{4} - 168552362t^{3} - 390494016t^{2} - 5861800t - 28512 \right)D^{6} + 2 \left( 2t + 1 \right) \left( 15864877204179214848t^{21} + 54055957137778568192t^{20} + 22154591625979344512t^{19} - 91766217017505971456t^{18} - 172094165234556774592t^{17} - 148763363430912562112t^{16} - 73066501466396692288t^{15} - 17200626385511227872t^{14} + 2096352720903315256t^{13} + 3022847726635182992t^{12} + 1020474339943887148t^{11} + 172395284633749928t^{10} + 14944357740088600t^{9} + 916009624385304t^{8} + 165336379086788t^{7} + 30733549414744t^{6} + 2518607919815t^{5} + 66986655199t^{4} + 4994921t^{3} + 763243232t^{2} + 8678652t + 28512 \right)D^{5} + \left( 528829240139307161600t^{22} + 2006599698966831201280t^{21} + 1264737488604788540864t^{20} - 3266532793834367362496t^{19} - 7473449343945980828000t^{18} - 7493859460707146005456t^{17} - 4517555078523667510464t^{16} - 1709086919039806806528t^{15} - 359708567547107448576t^{14} - 7304686678249923792t^{13} + 18897812173592111576t^{12} + 5366516105928172564t^{11} + 635675714511837132t^{10} + 18390067044939216t^{9} - 3662406332567712t^{8} - 266968605676128t^{7} + 77777829349960t^{6} + 11157804067970t^{5} + 254958383147t^{4} + 3132876526t^{3} - 1419028792t^{2} - 13938680t - 28512 \right)D^{4} + 2 t \left( 1110541404292545039360t^{21} + 4115387105432867345920t^{20} + 2051053593365155884480t^{19} - 7731815539364921880128t^{18} - 15953388205471596194144t^{17} - 15171702204741921400400t^{16} - 8781784672877970959168t^{15} - 3280531185817663564672t^{14} - 767724987428447059280t^{13} - 91384345570344835240t^{12} + 2473707914244824144t^{11} + 2449631967669212288t^{10} + 330693611465243468t^{9} + 30669654570648664t^{8} + 6107972272853696t^{7} + 1458811177834852t^{6} + 222376216254598t^{5} + 13844620781167t^{4} + 302118499672t^{3} + 3834425151t^{2} + 22248172t + 14580 \right)D^{3} + 2 t ^{2} \left( 2453767674246385229824t^{20} + 8926580779855931884544t^{19} + 3525819350627675682208t^{18} - 18455907586098081264928t^{17} - 35578396922565697779344t^{16} - 32510861281927284237624t^{15} - 18128043774441355447552t^{14} - 6572040755905618278496t^{13} - 1552180812988114021648t^{12} - 223860818907920872800t^{11} - 14609916206626145572t^{10} + 869086412577002206t^{9} + 352504027886170816t^{8} + 75071193931932288t^{7} + 14579148113486320t^{6} + 2534664717954868t^{5} + 296928173373070t^{4} + 13851066564781t^{3} + 257989514898t^{2} + 3757857318t + 20660112 \right)D^{2} + 8 t ^{2} \left( 666324842575527023616t^{20} + 2390902054534453676032t^{19} + 759523190100584804976t^{18} - 5270475274615855491632t^{17} - 9711720953855032885464t^{16} - 8617804617866286596884t^{15} - 4657141846117097756208t^{14} - 1634332691293654485216t^{13} - 378698841446272050600t^{12} - 56875466536820945084t^{11} - 4918705886009607138t^{10} - 26213414678014105t^{9} + 82239658098390864t^{8} + 21183008229406136t^{7} + 3573505994956566t^{6} + 526464195113235t^{5} + 51280240628892t^{4} + 1890650850318t^{3} + 33237125568t^{2} + 448943376t + 2223936 \right)D + 48 t ^{2} \left( 45328220583369185280t^{20} + 161123945231201576960t^{19} + 42635963030218179552t^{18} - 370021420005520593760t^{17} - 662553228818375635248t^{16} - 576292442078798911016t^{15} - 304282353905484160752t^{14} - 103904703478576286544t^{13} - 23507557031456724984t^{12} - 3534080486159657596t^{11} - 325204882852941156t^{10} - 3934419291327842t^{9} + 5590763706963120t^{8} + 1409231964174832t^{7} + 214813876369782t^{6} + 28582884773385t^{5} + 2411736073656t^{4} + 73886636040t^{3} + 1309917960t^{2} + 14551272t + 57024 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 13.08608 λ ≈ 13.08608; multiplicity 1; Jordan blocks 1 λ ≈ -4.127793 - 6.898065i; multiplicity 1; Jordan blocks 1 λ ≈ -4.127793 + 6.898065i; multiplicity 1; Jordan blocks 1 λ ≈ -3.561673; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ 2; multiplicity 1; Jordan blocks 1 λ ≈ -0.6344102 - 1.350313i; multiplicity 1; Jordan blocks 1 λ ≈ -0.6344102 + 1.350313i; multiplicity 1; Jordan blocks 1 2
Sector dimension
9
Spectral radius
$13.0860795$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.08608$11
$-4.127793 - 6.898065\mathrm{i}$11
$-4.127793 + 6.898065\mathrm{i}$11
$-3.561673$11
$-2$22
$2$11
$-0.6344102 - 1.350313\mathrm{i}$11
$-0.6344102 + 1.350313\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right) \left(\lambda^{6} - 60 \lambda^{4} - 1080 \lambda^{3} - 4416 \lambda^{2} - 6048 \lambda - 6704\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right) \left(\lambda^{6} - 60 \lambda^{4} - 1080 \lambda^{3} - 4416 \lambda^{2} - 6048 \lambda - 6704\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 24 & 6 & 24 & 64 & 32 & 40 \\ 1 & 0 & 0 & 4 & 0 & 8 & 24 & 12 & 0 \\ 2 & 3 & 0 & 4 & 2 & 4 & 36 & 12 & 64 \\ 0 & 2 & 1 & 0 & 1 & 0 & 4 & 0 & 24 \\ 0 & 1 & 0 & 0 & -2 & 0 & 8 & 4 & 0 \\ 0 & 0 & 2 & 3 & 1 & 0 & 4 & 2 & 12 \\ 0 & 0 & 0 & 2 & 0 & 5 & 6 & 3 & 8 \\ 0 & 0 & 0 & 1 & 2 & -8 & -12 & -6 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$