Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-513-105-a

·

Blow-up of $\mathbb P^4$ along a line.

  • $\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a line, the intersection of three general hyperplanes.

  • $\pi_2\colon X\to\mathbb P^2$ is the $\mathbb P^2$-bundle $\mathbb P_{\mathbb P^2}(\mathcal O^{\oplus 2}\oplus\mathcal O(-1))$.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

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^(w2 + w3)
Bundle
Q_1 box O_2(1)
ZL62 label
2040.l
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 105
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 513
$\chi(X, \mathrm{T}_X)$ = 18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 513
$\mathrm{c}_1^2\mathrm{c}_2$ = 222
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 9
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 3 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 18 0 18
$\mathrm{HH}^{ 2 }$ = 93 0 0 93
$\mathrm{HH}^{ 3 }$ = 172 0 0 0 172
$\mathrm{HH}^{ 4 }$ = 105 0 0 0 0 105
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 100t + 311t^{2} + 100t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(5)
Coates et al. (2020) Obro(4,118)
Coates–Kasprzyk–Prince (2015) CKP(8)

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( 69683t^{9} + 72900t^{8} + 27000t^{7} - 2187t^{6} + 4077t^{5} - 16t^{4} + 81t^{3} - 8t^{2} - 1 \right) \left( 20410953487500t^{11} - 77441303458800t^{10} + 70793124774750t^{9} + 56078054423500t^{8} - 17047426761990t^{7} - 34866033862020t^{6} - 1667344562937t^{5} + 109505543250t^{4} + 47392464960t^{3} - 15093913302t^{2} + 888569352t - 19699200 \right)D^{6} + \left( 29868225909258712500t^{20} - 93424237019171578800t^{19} + 19006544490089937750t^{18} + 162165485214011374500t^{17} + 82844841500329664250t^{16} - 68638767839935248510t^{15} - 66333753625828362867t^{14} - 17202862544136623610t^{13} - 139019263924310140t^{12} - 1248337192807243395t^{11} - 75748614524751013t^{10} - 8615677596760746t^{9} + 3313525259903481t^{8} - 1083423357161838t^{7} - 271849262453520t^{6} - 24162289071237t^{5} + 1625148027816t^{4} + 297656508672t^{3} - 89933105412t^{2} + 4442846760t - 78796800 \right)D^{5} + \left( 248901882577155937500t^{20} - 854189817732260226000t^{19} + 337831873382704295250t^{18} + 1445820491555035295000t^{17} + 528632512191216862110t^{16} - 703074323150864029350t^{15} - 560617705418097542085t^{14} - 133020944864950342920t^{13} + 3980259840826950t^{12} - 3684757045022928183t^{11} - 372698127241704615t^{10} - 36202830383578590t^{9} + 17748167018943764t^{8} - 4845519739597374t^{7} + 1242768593636220t^{6} + 14797454055894t^{5} + 1338740594274t^{4} - 661280145696t^{3} + 135495403398t^{2} - 4792889232t + 78796800 \right)D^{4} + t ^{2} \left( 1045387906824054937500t^{18} - 3837257114964733278000t^{17} + 2086439054464406486250t^{16} + 6341193123425234137500t^{15} + 1659808857266362303470t^{14} - 3480448992583611257850t^{13} - 2315623188621431229885t^{12} - 490460957890803772590t^{11} - 515139945655764060t^{10} - 1716923910575613681t^{9} - 1237620916056571175t^{8} - 221222979193356540t^{7} + 68421134904935691t^{6} - 16410857911252902t^{5} + 1020605810031240t^{4} - 45644719041738t^{3} + 2026352577000t^{2} - 91212106752t - 44236800 \right)D^{3} + t ^{3} \left( 2309809470316007100000t^{17} - 8900555042463412179600t^{16} + 5831977183582765774500t^{15} + 14409721594880559992000t^{14} + 2663392616510741420760t^{13} - 8691613928300237779140t^{12} - 4945361173356355431534t^{11} - 928808932602918572760t^{10} - 6984742597226485470t^{9} + 8576910450871862910t^{8} - 2680850556539508472t^{7} - 696971495734422192t^{6} + 117446040269500793t^{5} - 27062267808981096t^{4} + 1655189069126784t^{3} - 53119873011780t^{2} + 877614691176t - 89951275008 \right)D^{2} + 18 t ^{3} \left( 139385054243207325000t^{17} - 555766079051476616400t^{16} + 408831756627444777000t^{15} + 885431456829226411000t^{14} + 113992002611330125860t^{13} - 571962965683348481880t^{12} - 286956265096662652296t^{11} - 48122047910318582100t^{10} - 797395362724741410t^{9} + 743130965864066652t^{8} - 175430780832443634t^{7} - 49895274629799648t^{6} + 5871402878747499t^{5} - 1264930312958244t^{4} + 78062474619360t^{3} - 2294147287296t^{2} - 16236585432t - 2535287040 \right)D + 720 t ^{3} \left( 1422296471869462500t^{17} - 5799746570659903800t^{16} + 4576068693399079800t^{15} + 9137013601688775500t^{14} + 829750623310149120t^{13} - 6176533044667022160t^{12} - 2830894369724346282t^{11} - 432273111800054130t^{10} - 11106455090427045t^{9} + 7837242198016044t^{8} - 1948527799552953t^{7} - 552969572003856t^{6} + 52631320552902t^{5} - 10564665405948t^{4} + 661085546436t^{3} - 20455809984t^{2} - 363505644t - 13296960 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 5.704617 λ ≈ 5.704617; multiplicity 1; Jordan blocks 1 λ ≈ -4.025264 - 3.704723i; multiplicity 1; Jordan blocks 1 λ ≈ -4.025264 + 3.704723i; multiplicity 1; Jordan blocks 1 λ ≈ 0.5214156 - 4.707422i; multiplicity 1; Jordan blocks 1 λ ≈ 0.5214156 + 4.707422i; multiplicity 1; Jordan blocks 1 λ ≈ 1.703214 - 2.891028i; multiplicity 1; Jordan blocks 1 λ ≈ 1.703214 + 2.891028i; multiplicity 1; Jordan blocks 1 λ ≈ -1.051675 - 0.7141921i; multiplicity 1; Jordan blocks 1 λ ≈ -1.051675 + 0.7141921i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$5.70461696$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$5.704617$11
$-4.025264 - 3.704723\mathrm{i}$11
$-4.025264 + 3.704723\mathrm{i}$11
$0.5214156 - 4.707422\mathrm{i}$11
$0.5214156 + 4.707422\mathrm{i}$11
$1.703214 - 2.891028\mathrm{i}$11
$1.703214 + 2.891028\mathrm{i}$11
$-1.051675 - 0.7141921\mathrm{i}$11
$-1.051675 + 0.7141921\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\lambda^{9} + 8 \lambda^{7} - 81 \lambda^{6} + 16 \lambda^{5} - 4077 \lambda^{4} + 2187 \lambda^{3} - 27000 \lambda^{2} - 72900 \lambda - 69683\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{9} + 8 \lambda^{7} - 81 \lambda^{6} + 16 \lambda^{5} - 4077 \lambda^{4} + 2187 \lambda^{3} - 27000 \lambda^{2} - 72900 \lambda - 69683\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 5 \\ 2 & 0 & 0 & 0 & -2 & 0 & 3 & 0 & 0 \\ 3 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 2 & 0 & 0 & 0 & 0 & -2 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 & 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 3 & 0\end{pmatrix}\notag\end{equation}$$