Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-208-48-a

·

Blow-up of a smooth quadric fourfold along a del Pezzo surface of degree $2$.

  • $\pi_1\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a del Pezzo surface of degree $2$, embedded as a surface of degree $6$ (the intersection of $Q^4$ with a Segre threefold $\mathbb P^1\times\mathbb P^2\subset\mathbb P^5$).

  • $\pi_2\colon X\to\mathbb P^2$ is a flat quadric surface fibration, with smooth fibers off a smooth plane quartic and quadric cones over it.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ2P2 × Gr(2, 4)
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_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w1 + w4)
Bundle
2 copies of O_1(1) box O_2(1)
ZL62 label
2031.bb
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 48
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 208
$\chi(X, \mathrm{T}_X)$ = -9

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 208
$\mathrm{c}_1^2\mathrm{c}_2$ = 148
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 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 + 9 0 x0 + 9 x1
$\mathrm{HH}^{ 3 }$ = x1 + 42 0 0 x1 - 3 45
$\mathrm{HH}^{ 4 }$ = 48 0 0 0 0 48
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 43t + 120t^{2} + 43t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(247)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(315)

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( 11t - 1 \right) \left( t + 1 \right) ^{2} \left( 4t^{2} + 1 \right) \left( 148t^{3} + 92t^{2} + 13t + 1 \right) \left( 999874731504t^{13} + 3216350515840t^{12} + 3317654760280t^{11} + 1468616782660t^{10} + 1702066074527t^{9} + 2256744877731t^{8} + 1237125182575t^{7} + 295810602033t^{6} + 29548036917t^{5} + 2431682431t^{4} + 153611951t^{3} + 74615627t^{2} + 725136t + 9568 \right)D^{6} + \left( t + 1 \right) \left( 110690132276418816t^{20} + 480530772265699584t^{19} + 809319207063254272t^{18} + 732639138440530048t^{17} + 601634446332993440t^{16} + 672876781149194848t^{15} + 642481471806443952t^{14} + 412488070702694920t^{13} + 182930763375836685t^{12} + 58608654621752197t^{11} + 13528020566488176t^{10} + 2167491599442076t^{9} + 275817429618990t^{8} + 40317413057970t^{7} + 6448499835234t^{6} + 813062221246t^{5} + 33692700127t^{4} + 715149583t^{3} + 295893372t^{2} + 2079728t + 19136 \right)D^{5} + \left( 748786188928715520t^{21} + 3768904140738182336t^{20} + 7559062405315767808t^{19} + 8125676107805708800t^{18} + 6796032016372758032t^{17} + 7383617905000681292t^{16} + 8070397066928188788t^{15} + 6120959428637234806t^{14} + 3045268931513830163t^{13} + 1022063467961299989t^{12} + 240214585680269347t^{11} + 41183021119295523t^{10} + 5642158652518158t^{9} + 765236290349228t^{8} + 97114155060136t^{7} + 13940546847534t^{6} + 1569201339207t^{5} + 15161684827t^{4} + 193616457t^{3} - 290653007t^{2} - 2181856t - 9568 \right)D^{4} + t \left( 2571917779363848960t^{20} + 12357666159161727360t^{19} + 23080708993993043712t^{18} + 22626389650220025312t^{17} + 19105462126218207264t^{16} + 23732725063464438496t^{15} + 26294349060946983664t^{14} + 18619643811891619052t^{13} + 8291740433332431777t^{12} + 2392015084816015226t^{11} + 467988011732655909t^{10} + 68736808367281322t^{9} + 10255939856105788t^{8} + 2011609098482058t^{7} + 322377350554834t^{6} + 43859634977802t^{5} + 3550536171409t^{4} + 71127313538t^{3} + 1320402047t^{2} + 12199614t + 4416 \right)D^{3} + 2 t ^{2} \left( 2357048699062565376t^{19} + 10934504497435662880t^{18} + 19272546948161313664t^{17} + 17468317959072799984t^{16} + 15310183106074512824t^{15} + 21226676324363243270t^{14} + 23252506753860223434t^{13} + 15416694506382116387t^{12} + 6255931768796490377t^{11} + 1608039120053620987t^{10} + 280858508587041882t^{9} + 40514495710693463t^{8} + 7308609549206723t^{7} + 1592813684221421t^{6} + 241168712283493t^{5} + 27939905984283t^{4} + 1726515817997t^{3} + 32784119083t^{2} + 710735540t + 4780152 \right)D^{2} + 4 t ^{2} \left( 1087367770009526016t^{19} + 4916678744803029600t^{18} + 8283086145848913344t^{17} + 7037598356784314608t^{16} + 6480648698535541064t^{15} + 9745097593468350906t^{14} + 10465189460278030858t^{13} + 6557855236318962633t^{12} + 2466428737755852838t^{11} + 582167535463472190t^{10} + 96640185825451965t^{9} + 15067499362566571t^{8} + 3177413203109036t^{7} + 677956986638522t^{6} + 94281547901145t^{5} + 9399032737019t^{4} + 454579169803t^{3} + 9280599536t^{2} + 194987616t + 956800 \right)D + 16 t ^{2} \left( 97667763773310720t^{19} + 433766631073257504t^{18} + 706805196367215264t^{17} + 570946698102720752t^{16} + 551924207178659520t^{15} + 877903276309916982t^{14} + 924773252780184846t^{13} + 554457188799795053t^{12} + 196543442122266080t^{11} + 43658137603702770t^{10} + 7182136784160493t^{9} + 1236650660831687t^{8} + 284724450145852t^{7} + 57664042527368t^{6} + 7414472833499t^{5} + 650791717197t^{4} + 25495746171t^{3} + 594365640t^{2} + 11117280t + 38272 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 4)

|λ| = 11 λ ≈ 11; multiplicity 1; Jordan blocks 1 λ ≈ -5.421453 - 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -5.421453 + 6.262482i; multiplicity 1; Jordan blocks 1 λ ≈ -2.157094; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2 2
Sector dimension
10
Spectral radius
$11$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11$11
$-5.421453 - 6.262482\mathrm{i}$11
$-5.421453 + 6.262482\mathrm{i}$11
$-2.157094$11
$0 - 2\mathrm{i}$11
$0 + 2\mathrm{i}$11
$-1$22
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda - 11\right) \left(\lambda^{2} + 4\right) \left(\lambda^{3} + 13 \lambda^{2} + 92 \lambda + 148\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda - 11\right) \left(\lambda^{2} + 4\right) \left(\lambda^{3} + 13 \lambda^{2} + 92 \lambda + 148\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 18 & 6 & 9 & 9 & 8 & 8 & 20 \\ 1 & -1 & 0 & 4 & 0 & 4 & 4 & 6 & 6 & 0 \\ 2 & 2 & 0 & 0 & 0 & 2 & 2 & 15 & 12 & 8 \\ 0 & 2 & 1 & -1 & 0 & 0 & 0 & 2 & 0 & 6 \\ 0 & 1 & 0 & 0 & -1 & 0 & 0 & 4 & 4 & 0 \\ 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 4 & 0 & 5 & 5 & 3 & 4 & 0 \\ 0 & 0 & 0 & 1 & 2 & -3 & -3 & -3 & -4 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$