Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-224-51-a

·

Blow-up of a cubic fourfold singular along a line.

  • $\pi_1\colon X\to Y$ is the blow-up of a cubic fourfold $Y\subset\mathbb P^5$ singular (with multiplicity $2$) along a line, along that line; the exceptional divisor is a quadric surface fibration over the line.

  • $\pi_2\colon X\to\mathbb P^3$ resolves the projection of the cubic fourfold from the singular line; its fibers are lines, except over $12$ points, where they are planes.

Varieties in this family can be blown down to

  • A cubic fourfold singular along a line
Defining data
Ambient
ℙ3P3 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w3 + w4) + E^(2w1 + w4)
Bundle
Q_1 box O_2(1) + O_1(2) box O_2(1)
ZL62 label
3050.wDx327
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(3;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 224
$\chi(X, \mathrm{T}_X)$ = -12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 224
$\mathrm{c}_1^2\mathrm{c}_2$ = 152
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 32
$\mathrm{c}_4$ = 20
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 14 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 + 12 0 x0 + 12 x1
$\mathrm{HH}^{ 3 }$ = x1 + 44 0 0 x1 - 12 56
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 46t + 130t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(128)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(170)

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( 985t^{8} + 432t^{7} - 1228t^{6} - 16360t^{5} - 6168t^{4} - 933t^{3} + 8t + 1 \right) \left( 3022705471664160t^{13} + 4428037855298160t^{12} + 3562336068927552t^{11} + 1984842721333284t^{10} + 1364179686279918t^{9} + 639216802530724t^{8} + 112291971803496t^{7} + 1363193919679t^{6} - 125211299954t^{5} + 401781984248t^{4} + 39618004316t^{3} + 2153544236t^{2} - 2928662t + 389088 \right)D^{6} + 2 \left( 25307601561508179600t^{21} + 47742312551651149680t^{20} + 30021475802995055520t^{19} - 119724136554136730520t^{18} - 229741494571904767029t^{17} - 229750363798020069556t^{16} - 165847101691817382408t^{15} - 128404846009035424756t^{14} - 74553238741406834865t^{13} - 22736544681240369915t^{12} - 3237286586790963598t^{11} - 231790427701494556t^{10} - 68375680877601722t^{9} - 22428409204580035t^{8} - 3277529850478376t^{7} - 212619194322038t^{6} - 10752916813113t^{5} - 1368321246089t^{4} - 100860106310t^{3} - 4318803120t^{2} + 7505697t - 389088 \right)D^{5} + \left( 342396962302757724000t^{21} + 641414582680370772240t^{20} + 522673190022672819360t^{19} - 221209942236830627556t^{18} - 716060015376415866462t^{17} - 873821199915540635996t^{16} - 764766596165667366156t^{15} - 703185180817215465375t^{14} - 418753316473125933474t^{13} - 109256702763046744962t^{12} - 7356620080791926972t^{11} + 697530059467727446t^{10} - 322966369308628989t^{9} - 120182890490008672t^{8} - 16441767046286324t^{7} - 1439461088363575t^{6} - 71476589608818t^{5} + 1366101699374t^{4} + 157766422452t^{3} + 8976678668t^{2} + 10788926t + 389088 \right)D^{4} + 2 t \left( 588029565693866526000t^{20} + 1095622156509359958000t^{19} + 1009843688404864759680t^{18} + 400572716440134460560t^{17} + 36337874959674004113t^{16} - 285795099490029892432t^{15} - 467088483837966875268t^{14} - 511422202639211631394t^{13} - 301758573219343462671t^{12} - 65488452298598514195t^{11} + 2223738560484233466t^{10} + 1608303430459882493t^{9} - 421880556938979744t^{8} - 135736192640514148t^{7} - 15817840655740684t^{6} - 1152894263408848t^{5} - 41108668362359t^{4} - 331833733949t^{3} - 16088088922t^{2} - 117735448t - 22581 \right)D^{3} + 4 t ^{2} \left( 538903045015644765600t^{19} + 999955582245187027920t^{18} + 985117726278707790600t^{17} + 614634777038989576644t^{16} + 406791163138025815476t^{15} + 119454469419609956920t^{14} - 139850279253730288845t^{13} - 205259022163600484421t^{12} - 116262739780647373656t^{11} - 19470961883066617028t^{10} + 4325369431553447213t^{9} + 1196655215941675475t^{8} - 234985209169612963t^{7} - 67537415109572050t^{6} - 7346309424532033t^{5} - 544624361573276t^{4} - 18425746103978t^{3} - 200115458016t^{2} - 9745798888t - 44672768 \right)D^{2} + 8 t ^{2} \left( 248609968280697999600t^{19} + 459874999880216790480t^{18} + 471261635643506698440t^{17} + 329306973057772033200t^{16} + 251606380240095530283t^{15} + 124561678535246648804t^{14} - 8648567484680742411t^{13} - 43437621093273831778t^{12} - 23395955123141740215t^{11} - 2761918158489508637t^{10} + 1654253867207043025t^{9} + 388230684497379552t^{8} - 62558539670081591t^{7} - 16138576982734874t^{6} - 1755935699164202t^{5} - 136879558404812t^{4} - 4518762816628t^{3} - 72338540620t^{2} - 2968917908t - 9078720 \right)D + 48 t ^{2} \left( 14886824447945988000t^{19} + 27475512488571042720t^{18} + 28853528903418549600t^{17} + 21046226480322622032t^{16} + 16837485246428314446t^{15} + 9412247830637696296t^{14} + 1074001926854361786t^{13} - 1269509034514510340t^{12} - 671852853657411380t^{11} - 57079085383990978t^{10} + 67781271786460941t^{9} + 15818604903178784t^{8} - 2058488861042942t^{7} - 467823884689824t^{6} - 55923277111602t^{5} - 4750185311938t^{4} - 157739636866t^{3} - 3636128812t^{2} - 122231918t - 259392 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ5P5

|λ| = 9.844861 λ ≈ 9.844861; multiplicity 1; Jordan blocks 1 λ ≈ -5.206832 - 7.040612i; multiplicity 1; Jordan blocks 1 λ ≈ -5.206832 + 7.040612i; multiplicity 1; Jordan blocks 1 λ ≈ -3.674411 - 2.740367i; multiplicity 1; Jordan blocks 1 λ ≈ -3.674411 + 2.740367i; multiplicity 1; Jordan blocks 1 λ ≈ -0.2276963 - 0.3385769i; multiplicity 1; Jordan blocks 1 λ ≈ -0.2276963 + 0.3385769i; multiplicity 1; Jordan blocks 1 λ ≈ 0.3730185; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$9.84486091$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.844861$11
$-5.206832 - 7.040612\mathrm{i}$11
$-5.206832 + 7.040612\mathrm{i}$11
$-3.674411 - 2.740367\mathrm{i}$11
$-3.674411 + 2.740367\mathrm{i}$11
$-0.2276963 - 0.3385769\mathrm{i}$11
$-0.2276963 + 0.3385769\mathrm{i}$11
$0.3730185$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{8} + 8 \lambda^{7} - 933 \lambda^{5} - 6168 \lambda^{4} - 16360 \lambda^{3} - 1228 \lambda^{2} + 432 \lambda + 985\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{8} + 8 \lambda^{7} - 933 \lambda^{5} - 6168 \lambda^{4} - 16360 \lambda^{3} - 1228 \lambda^{2} + 432 \lambda + 985\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 18 & 12 & 18 & 0 & 0 & 15 \\ 1 & -2 & 0 & 2 & 4 & 6 & 18 & 18 & 0 \\ 2 & 2 & 0 & 0 & -4 & 0 & 27 & 15 & 0 \\ 0 & 2 & 1 & -2 & -1 & 0 & 0 & 4 & 9 \\ 0 & 1 & 0 & 0 & -2 & 0 & 6 & 2 & 0 \\ 0 & 0 & 2 & 2 & 3 & 0 & 0 & -4 & 9 \\ 0 & 0 & 0 & 2 & -\frac{1}{6} & 3 & 0 & 2 & -1 \\ 0 & 0 & 0 & 1 & \frac{5}{2} & 0 & 0 & -2 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{7}{3} & 0\end{pmatrix}\notag\end{equation}$$