Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-352-75-a

·

Small resolution of a determinantal fourfold of degree $7$ in $\operatorname{Gr}(2,6)$.

  • $\pi_1\colon X\to\mathrm{Gr}(2,6)$ maps birationally onto a fourfold $D$ of degree $7$; it contracts a plane with normal bundle $\mathcal O(-1)^{\oplus 2}$ to the unique singular point of $D$, locally the cone over $\mathbb P^2\times\mathbb P^1$.

  • $\pi_2\colon X\to\mathbb P^3$ is a fibration with general fiber $\mathbb P^1$ and exactly four fibers isomorphic to $\mathbb P^2$.

Defining data
Ambient
ℙ3P3 × Gr(2, 6)
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_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w1 + w5) + E^(w3 + w4)
Bundle
O_1(1) box O_2(1) + Q_1 box U_2^v
ZL62 label
3051.wDw3
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 75
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 352
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 352
$\mathrm{c}_1^2\mathrm{c}_2$ = 184
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 6 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 - 4 0 x0 - 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 60 0 0 x1 - 44 104
$\mathrm{HH}^{ 4 }$ = 75 0 0 0 0 75
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 70t + 210t^{2} + 70t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(70)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(114)

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} - 17056t^{7} + 17564t^{6} - 6024t^{5} + 312t^{4} - 201t^{3} - 16t^{2} - t + 1 \right) \left( 13435320325320t^{12} - 32967523904760t^{11} + 45574242166728t^{10} + 24458050174138t^{9} - 57104517714474t^{8} + 32238545263130t^{7} - 7177013666289t^{6} + 2374065328383t^{5} - 861709763826t^{4} + 158251900339t^{3} - 6510821025t^{2} + 239207744t + 506880 \right)D^{6} + \left( 224974438847483400t^{20} - 3563500903923947760t^{19} + 10848840865969810680t^{18} - 17372978274335857720t^{17} + 2145184878290203454t^{16} + 20476907834854921748t^{15} - 24246648339591348471t^{14} + 13441909495067347592t^{13} - 4749935068228843950t^{12} + 1529709149480802312t^{11} - 520239272412614063t^{10} + 130666395651297381t^{9} - 19238681984967510t^{8} + 1673624271239385t^{7} - 148189028586261t^{6} - 16637302320719t^{5} + 6278401409588t^{4} - 972248464595t^{3} + 32513554725t^{2} - 957337856t - 1520640 \right)D^{5} + \left( 1521885909850623000t^{20} - 19530246737347793640t^{19} + 56479805659545515280t^{18} - 85147753537188281978t^{17} - 12908775212928538862t^{16} + 132088747220624561690t^{15} - 135598210338830420771t^{14} + 68761090850586256381t^{13} - 24300645000633352692t^{12} + 8679864022746686151t^{11} - 3012655938935794369t^{10} + 737591363095961934t^{9} - 103025954624352913t^{8} + 8374263855707490t^{7} - 885272162330037t^{6} + 70448631443375t^{5} - 10926942053633t^{4} + 1328487198889t^{3} - 32754394247t^{2} + 981041696t + 1013760 \right)D^{4} + t \left( 5227347255573879000t^{19} - 55243950698617614000t^{18} + 152336797465074378840t^{17} - 212851573554945739600t^{16} - 93058019580267423782t^{15} + 409600256204870564304t^{14} - 380494312972176874533t^{13} + 178755585093208604800t^{12} - 65707014763975748862t^{11} + 26715443258427917692t^{10} - 9439791641592054949t^{9} + 2190183201345520798t^{8} - 277317839887177662t^{7} + 16065029842875502t^{6} - 1149216449531961t^{5} + 53407604481200t^{4} - 2838385521602t^{3} - 34627405292t^{2} - 130078845t + 1440 \right)D^{3} + 2 t ^{2} \left( 4790632168399352400t^{18} - 42586620461098669440t^{17} + 112314676911161691060t^{16} - 143074507280729117836t^{15} - 105612751185096178188t^{14} + 330591321248581569072t^{13} - 284699741449473136690t^{12} + 125624381925471488734t^{11} - 49401438852818066748t^{10} + 22613265676813523854t^{9} - 8081539273376814075t^{8} + 1784362970376878866t^{7} - 213642451736842322t^{6} + 10933167788510938t^{5} - 546944635260465t^{4} + 15988984675916t^{3} - 1699972498164t^{2} - 8920387712t - 36315136 \right)D^{2} + 4 t ^{2} \left( 2210043016913513400t^{18} - 16944917122319643360t^{17} + 42883186563443835540t^{16} - 49042886446306419280t^{15} - 52208832998505330894t^{14} + 133703425674531865852t^{13} - 108299526031064435757t^{12} + 45468916739782609120t^{11} - 19455421991624406498t^{10} + 9793070500094953876t^{9} - 3521124653682855636t^{8} + 744741766937849644t^{7} - 85494874291339956t^{6} + 4203101752979740t^{5} - 157321169649987t^{4} - 802638141184t^{3} - 479245011516t^{2} - 1614314880t - 4055040 \right)D + 48 t ^{3} \left( 66168952602201000t^{17} - 450458470786669920t^{16} + 1099292967093810420t^{15} - 1118461051278825784t^{14} - 1586327892344145390t^{13} + 3551467728488800324t^{12} - 2734468432581670023t^{11} + 1104807678322915816t^{10} - 516353437049023242t^{9} + 278841602210673148t^{8} - 100527222136074495t^{7} + 20538540201856672t^{6} - 2283954305396529t^{5} + 112688795695954t^{4} - 3636496934430t^{3} - 141871386526t^{2} - 8745006267t - 34642080 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 6)

|λ| = 7.393565 λ ≈ 7.393565; multiplicity 1; Jordan blocks 1 λ ≈ -5.040373 - 4.263873i; multiplicity 1; Jordan blocks 1 λ ≈ -5.040373 + 4.263873i; multiplicity 1; Jordan blocks 1 λ ≈ 0.4006149 - 4.439574i; multiplicity 1; Jordan blocks 1 λ ≈ 0.4006149 + 4.439574i; multiplicity 1; Jordan blocks 1 λ ≈ 1.41219 - 0.7099316i; multiplicity 1; Jordan blocks 1 λ ≈ 1.41219 + 0.7099316i; multiplicity 1; Jordan blocks 1 λ ≈ 0.06157288; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$7.39356507$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.393565$11
$-5.040373 - 4.263873\mathrm{i}$11
$-5.040373 + 4.263873\mathrm{i}$11
$0.4006149 - 4.439574\mathrm{i}$11
$0.4006149 + 4.439574\mathrm{i}$11
$1.41219 - 0.7099316\mathrm{i}$11
$1.41219 + 0.7099316\mathrm{i}$11
$0.06157288$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{8} - \lambda^{7} - 16 \lambda^{6} - 201 \lambda^{5} + 312 \lambda^{4} - 6024 \lambda^{3} + 17564 \lambda^{2} - 17056 \lambda + 985\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{8} - \lambda^{7} - 16 \lambda^{6} - 201 \lambda^{5} + 312 \lambda^{4} - 6024 \lambda^{3} + 17564 \lambda^{2} - 17056 \lambda + 985\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 6 & 6 & 0 & 0 & 0 & 10 \\ 1 & 0 & 0 & 2 & 0 & 4 & -6 & -3 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 6 & 9 & 0 \\ 0 & 2 & 3 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 4 & 2 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 2 & -\frac{1}{4} & 3 & 0 & 0 & -1 \\ 0 & 0 & 0 & 3 & \frac{5}{2} & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$