Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-115-30-b

·

Blow-up of a cubic fourfold along a del Pezzo surface of degree $4$.

  • $\pi_1\colon X\to Y_3$ is the blow-up of a smooth cubic fourfold $Y_3\subset\mathbb P^5$ (1-243-55-a) containing a plane, along a quartic del Pezzo surface (a complete intersection of two quadrics in a hyperplane $\mathbb P^4$) meeting the plane in $4$ points.

  • $\pi_2\colon X\to Y_8$ is the blow-up of an intersection $Y_8$ of three quadrics in $\mathbb P^7$ singular along a line, along that line; the exceptional divisor is a quadric surface fibration over the line with $6$ singular (cone) fibers.

This is a Fano fourfold of K3 type, appearing as case C-8 in Bernardara–Fatighenti–Manivel–Tanturri.

Varieties in this family can be blown down to

  • 1-243-55-a (containing a plane)
  • A complete intersection of three quadrics in $\mathbb P^7$ singular along a line
Defining data
Ambient
Fl(1, 3, 8)
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_7ω1ω2ω3ω4ω5ω6ω7
Bundle
2 copies of E^(w7) + E^(w1 + w3) + E^(w2 + w3)
Bundle
2 copies of wedge^4 Q_1^v ot wedge^2 U_2^v ot U_3^v + O(1,1) + Sigma_(2,1) U_2^v ot Sym^2 U_3^v
ZL62 label
704.66w6w1
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 115
$\chi(X, \mathrm{T}_X)$ = -26

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 115
$\mathrm{c}_1^2\mathrm{c}_2$ = 118
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 14
$\mathrm{c}_4$ = 35
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 27 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 + 26 0 x0 + 26 x2
$\mathrm{HH}^{ 3 }$ = x1 + 18 0 0 x1 18
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 26 0 0 x1 - x2 - 4 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 63t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( t + 1 \right) ^{2} \left( 1369t^{4} + 3165t^{3} + 284t^{2} - 9t - 1 \right) \left( 4479101600t^{10} + 53444262710t^{9} + 126983358866t^{8} + 281286521826t^{7} + 260054067989t^{6} + 123559733224t^{5} + 47729859055t^{4} + 8742402590t^{3} + 27753407t^{2} + 1109762t + 4715 \right)D^{6} + 2 \left( t + 1 \right) \left( 36791340542400t^{15} + 544542978161735t^{14} + 2163518558493828t^{13} + 5539118514319442t^{12} + 9579646253959769t^{11} + 9379876639275221t^{10} + 5792875036648931t^{9} + 2635864491246943t^{8} + 815412208200972t^{7} + 126359299141515t^{6} + 4701627291174t^{5} + 202215494016t^{4} + 22627473615t^{3} + 51543485t^{2} + 1608063t + 4715 \right)D^{5} + \left( 355649625243200t^{16} + 5527362279425770t^{15} + 23090156717622128t^{14} + 64919353619816598t^{13} + 123375510007828439t^{12} + 142370743368908089t^{11} + 105343727203496236t^{10} + 56115145102827713t^{9} + 22036681958371815t^{8} + 5349717098748828t^{7} + 598836615311024t^{6} + 6723351970584t^{5} - 115801662755t^{4} - 46747371503t^{3} - 142625724t^{2} - 2405951t - 4715 \right)D^{4} + 2 t \left( 441496086508800t^{15} + 6797189474294645t^{14} + 25839955387725699t^{13} + 73944370322543484t^{12} + 136341084999521293t^{11} + 147081012517443656t^{10} + 103124911318455621t^{9} + 53561110158541527t^{8} + 19940004944093492t^{7} + 4279564634765245t^{6} + 379046576918087t^{5} - 140421578330t^{4} + 165025195527t^{3} + 2346348886t^{2} + 11861865t + 1271 \right)D^{3} + 4 t ^{2} \left( 295863696861800t^{14} + 4532102836696420t^{13} + 15912304433069638t^{12} + 47045401081363347t^{11} + 84351504795279624t^{10} + 86467201191432745t^{9} + 58462554715329689t^{8} + 29938586984285160t^{7} + 10529828211162777t^{6} + 2009569559313820t^{5} + 133529283760783t^{4} - 804363195615t^{3} + 89228535499t^{2} + 826241787t + 4586430 \right)D^{2} + 8 t ^{2} \left( 101176186491600t^{14} + 1545927971604245t^{13} + 5086148107120087t^{12} + 15600269870072897t^{11} + 27316012642241122t^{10} + 26974289446264963t^{9} + 17802183504913803t^{8} + 9046799290106418t^{7} + 3022393768689989t^{6} + 523298193683811t^{5} + 24799133212982t^{4} - 82337266139t^{3} + 23647908581t^{2} + 157927437t + 924140 \right)D + 16 t ^{2} \left( 13796752703400t^{14} + 210568275014080t^{13} + 657547867917816t^{12} + 2088665859579046t^{11} + 3589050242987098t^{10} + 3449696112657807t^{9} + 2240439551936939t^{8} + 1133608578586276t^{7} + 362861845494019t^{6} + 58346824219502t^{5} + 1883881303937t^{4} + 20567572518t^{3} + 2538718855t^{2} + 13038451t + 75440 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 8)

|λ| = 17.60162 λ ≈ 17.60162; multiplicity 1; Jordan blocks 1 λ ≈ -13.07529 - 1.213848i; multiplicity 1; Jordan blocks 1 λ ≈ -13.07529 + 1.213848i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ -0.4510465; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 3; Jordan blocks 1, 2 2 3
Sector dimension
9
Spectral radius
$17.6016223$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$17.60162$11
$-13.07529 - 1.213848\mathrm{i}$11
$-13.07529 + 1.213848\mathrm{i}$11
$-1$22
$-0.4510465$11
$0$31, 2

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 1\right)^{2} \left(\lambda^{4} + 9 \lambda^{3} - 284 \lambda^{2} - 3165 \lambda - 1369\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{4} + 9 \lambda^{3} - 284 \lambda^{2} - 3165 \lambda - 1369\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 36 & 48 & 18 & 416 & 448 & 480 \\ 1 & -1 & 2 & 24 & 32 & 8 & 24 & 0 & 240 \\ 1 & 2 & -2 & 24 & 16 & 36 & 66 & 120 & 96 \\ 0 & 1 & 1 & -3 & -2 & -3 & 16 & 16 & 0 \\ 0 & 1 & 0 & 2 & 0 & 5 & 20 & 32 & 0 \\ 0 & 0 & 1 & 2 & 4 & -2 & 8 & 0 & 36 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 & 2 & -\frac{3}{4} & \frac{1}{2} & -3 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & \frac{8}{3} & 0\end{pmatrix}\notag\end{equation}$$