Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-116-30-d

·

Blow-up of $\mathbb P^4$ along a smooth rational surface of degree $8$ and sectional genus $6$.

  • $\pi_1\colon X\to Y$ maps birationally onto a fourfold $Y\subset\mathbb P^8$ of degree $11$; it contracts five disjoint planes with normal bundle $\mathcal O(-1)^{\oplus 2}$ to the five singular points of $Y$, each a cone over the Segre embedding of $\mathbb P^1\times\mathbb P^2$.

  • $\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a smooth surface of degree $8$ and sectional genus $6$, the blow-up of $\mathbb P^2$ in sixteen points in special position.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ4P4 × ℙ8P8
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11node 12: fundamental weight ω_12ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12
Bundle
E^(w2 + w5) + E^(w4 + w5)
Bundle
Q_1^v(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
40800.w0Uw0D
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;9) \\ \det\mathcal{E} &\cong \mathcal{O}(4;8) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 116
$\chi(X, \mathrm{T}_X)$ = -19

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 116
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 124
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 18 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 + x2 + 19 0 x0 + 19 x2
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 15
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 19 0 0 x1 - x2 - 11 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 64t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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) \left( 73t^{7} - 562t^{6} + 752t^{5} + 4045t^{4} + 2512t^{3} + 180t^{2} - 2t - 1 \right) \left( 386134074510t^{13} + 1087195501380t^{12} - 5833162555805t^{11} + 1112040557625t^{10} - 1391423900768t^{9} - 5164767375223t^{8} - 2221603735668t^{7} - 4291139611645t^{6} - 1009854737431t^{5} + 233968003611t^{4} - 11472257917t^{3} - 3403087376t^{2} - 23345937t - 229100 \right)D^{6} + \left( 479192386466910t^{21} - 1026079422846750t^{20} - 14873247599936915t^{19} + 55736923790706525t^{18} + 10798574142913436t^{17} - 178924959488160289t^{16} - 40109663102463221t^{15} - 16966290602860146t^{14} - 254349011165877965t^{13} - 327330197915768676t^{12} - 381899250198294972t^{11} - 303268217601739144t^{10} - 89803077423399246t^{9} + 825532353069050t^{8} + 2439065395527961t^{7} - 445981750980068t^{6} - 87090063481982t^{5} - 1640263247233t^{4} - 84306545515t^{3} - 13786990493t^{2} - 68663211t - 458200 \right)D^{5} + \left( 3241595555511450t^{21} - 2571521189877150t^{20} - 99601183989738185t^{19} + 288819323445425310t^{18} - 63908536788202915t^{17} - 441781804918106210t^{16} + 455630438339749625t^{15} - 107088921137717356t^{14} - 918726097526561117t^{13} - 1083723194392094720t^{12} - 1527479401178677417t^{11} - 991663060940769882t^{10} - 152109168311367966t^{9} + 38037694064430734t^{8} + 4150582402787780t^{7} - 1638224730708975t^{6} - 222849602231713t^{5} - 8335603625728t^{4} - 56938743245t^{3} + 12513503357t^{2} + 70157639t + 229100 \right)D^{4} + t \left( 11134176038495850t^{20} + 2670948794139750t^{19} - 338788609171869485t^{18} + 821253132909929895t^{17} - 341611163891418380t^{16} - 379400321091695265t^{15} + 1563869143457603735t^{14} - 807954940266188166t^{13} - 1824703392351088361t^{12} - 2232508175103985284t^{11} - 3476993231504936990t^{10} - 1702326425039503032t^{9} - 51166502482714302t^{8} + 100938525473133554t^{7} - 1338465158755105t^{6} - 3249193454511368t^{5} - 361861837377504t^{4} - 6031678337189t^{3} - 97528292265t^{2} - 839329543t - 235025 \right)D^{3} + 2 t ^{2} \left( 10203979053001260t^{19} + 10456157998394700t^{18} - 307591023755993725t^{17} + 645843662236228725t^{16} - 342095046980538378t^{15} + 33721967263744741t^{14} + 1107882576583267971t^{13} - 914238301558158136t^{12} - 912119831738710752t^{11} - 1400227059365832623t^{10} - 2209647920700245258t^{9} - 814060285862323845t^{8} + 75322575668841836t^{7} + 57625939663985695t^{6} - 4213449869917613t^{5} - 1664457619959916t^{4} - 155481231764374t^{3} - 2403612408529t^{2} - 52032925763t - 320056296 \right)D^{2} + 4 t ^{2} \left( 4707360502351410t^{19} + 7595082592224750t^{18} - 140667721028545130t^{17} + 261314866385450670t^{16} - 160869943683303279t^{15} + 64044660080916301t^{14} + 389587468848513604t^{13} - 441312542079260155t^{12} - 193277999531899130t^{11} - 481757670695816573t^{10} - 735187645167139282t^{9} - 207625382495878560t^{8} + 48357125997206050t^{7} + 15535810747299164t^{6} - 1829151346736790t^{5} - 472291547205033t^{4} - 36243844538073t^{3} - 623881723944t^{2} - 13459336776t - 61032240 \right)D + 40 t ^{2} \left( 169126724635380t^{19} + 344846480602860t^{18} - 5016635384791740t^{17} + 8418760664334360t^{16} - 5804680725402067t^{15} + 2356439599357183t^{14} + 11197297600127508t^{13} - 15719916459546283t^{12} - 1722805945505832t^{11} - 13696881105794933t^{10} - 20000135527849744t^{9} - 4431757737886416t^{8} + 1782471211928784t^{7} + 323994814423528t^{6} - 51881288204332t^{5} - 11725434106959t^{4} - 721415518307t^{3} - 15190054632t^{2} - 282400510t - 916400 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ8P8

|λ| = 17.40653 λ ≈ 17.40653; multiplicity 1; Jordan blocks 1 λ ≈ -8.791686 - 6.934213i; multiplicity 1; Jordan blocks 1 λ ≈ -8.791686 + 6.934213i; multiplicity 1; Jordan blocks 1 λ ≈ -1.427414; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ -0.7180239; multiplicity 1; Jordan blocks 1 λ ≈ 0.1611379 - 0.08167666i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1611379 + 0.08167666i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$17.4065346$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$17.40653$11
$-8.791686 - 6.934213\mathrm{i}$11
$-8.791686 + 6.934213\mathrm{i}$11
$-1.427414$11
$-1$11
$-0.7180239$11
$0.1611379 - 0.08167666\mathrm{i}$11
$0.1611379 + 0.08167666\mathrm{i}$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda^{7} + 2 \lambda^{6} - 180 \lambda^{5} - 2512 \lambda^{4} - 4045 \lambda^{3} - 752 \lambda^{2} + 562 \lambda - 73\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda^{7} + 2 \lambda^{6} - 180 \lambda^{5} - 2512 \lambda^{4} - 4045 \lambda^{3} - 752 \lambda^{2} + 562 \lambda - 73\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 90 & 45 & 135 & 312 & 240 & 1350 \\ 1 & 0 & 4 & -4 & 6 & -14 & 156 & 96 & 40 \\ 1 & 0 & -1 & 28 & 18 & 38 & 108 & 72 & 280 \\ 0 & 1 & 1 & -1 & -3 & 0 & -16 & -8 & 30 \\ 0 & 1 & 0 & 4 & 1 & 8 & 4 & 8 & 90 \\ 0 & 0 & 1 & 0 & 2 & -1 & 36 & 24 & 30 \\ 0 & 0 & 0 & 1 & -\frac{1}{3} & \frac{5}{2} & -\frac{11}{3} & -\frac{4}{3} & \frac{50}{3} \\ 0 & 0 & 0 & 1 & \frac{23}{12} & -\frac{1}{2} & \frac{22}{3} & \frac{8}{3} & -\frac{25}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{9}{5} & \frac{6}{5} & 0\end{pmatrix}\notag\end{equation}$$