Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-432-90-a

·

Blow-up of a smooth quadric fourfold along a plane.

  • $\pi_1\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a plane.

  • $\pi_2\colon X\to\mathbb P^2$ is the $\mathbb P^2$-bundle $\mathbb P_{\mathbb P^2}(T_{\mathbb P^2}(-1)\oplus\mathcal O)$, resolving the linear projection of $Q^4\subset\mathbb P^5$ (1-512-105-a) from the plane.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3ω1ω2ω3
Bundle
E^(w1)
Bundle
O(1,0)
ZL62 label
32.0
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,0) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 90
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 432
$\chi(X, \mathrm{T}_X)$ = 12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 432
$\mathrm{c}_1^2\mathrm{c}_2$ = 204
$\mathrm{c}_2^2$ = 97
$\mathrm{c}_1\mathrm{c}_3$ = 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
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 12 0 x0 - 12 x1
$\mathrm{HH}^{ 3 }$ = x1 + 70 0 0 x1 - 66 136
$\mathrm{HH}^{ 4 }$ = 90 0 0 0 0 90
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 85t + 260t^{2} + 85t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(10)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(14)

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( 20707t^{9} + 3456t^{8} + 25081t^{7} + 1013t^{6} + 3445t^{5} + 1377t^{4} + 80t^{3} - 3t^{2} - 3t - 1 \right) \left( 18853426315848t^{13} + 309617457739928t^{12} + 410032747708400t^{11} + 22378541992384t^{10} - 55020125037422t^{9} + 3190913389592t^{8} + 40016044787818t^{7} - 930079349154t^{6} - 207817583278t^{5} + 303740521501t^{4} + 6458887080t^{3} + 1111701339t^{2} + 2408616t + 256608 \right)D^{6} + 2 \left( 4099177936583777628t^{22} + 71077573923081930904t^{21} + 110345252738998460684t^{20} + 73467456348084466940t^{19} + 65392953444038903335t^{18} + 8849979026205523196t^{17} + 5081980530940200374t^{16} + 2809498729500173593t^{15} + 8651637303542814543t^{14} + 29531914663018765t^{13} + 730475163726552580t^{12} + 259253554326636260t^{11} - 721613913008777t^{10} + 6071239032004514t^{9} + 2544676340011370t^{8} + 327365334240942t^{7} + 5500036440312t^{6} + 540053908198t^{5} + 917580233361t^{4} + 16231796280t^{3} + 2217480282t^{2} + 2843100t + 256608 \right)D^{5} + \left( 68319632276396293800t^{22} + 1225630358264901557360t^{21} + 1910848380447268297000t^{20} + 890674146720822063376t^{19} + 628693534560872629426t^{18} + 72019635132677475752t^{17} + 109219372052282874660t^{16} + 33424667580642492560t^{15} + 134752864442608468902t^{14} + 3067809286864501584t^{13} + 6069662437077097302t^{12} + 2191106982426132140t^{11} + 1093914821741940t^{10} + 83695654044178248t^{9} + 18075780108816025t^{8} + 243381479892723t^{7} + 106419148472275t^{6} + 4222907330352t^{5} - 2674928929363t^{4} - 27053886915t^{3} - 4457366019t^{2} - 4582008t - 256608 \right)D^{4} + t \left( 286942455560864433960t^{21} + 5282961121217255480480t^{20} + 8275308037564067860680t^{19} + 2777869038606510929992t^{18} + 1244331023976621790642t^{17} + 208415490181014225752t^{16} + 801091663063966357396t^{15} + 175670913942728291430t^{14} + 513595105482517978874t^{13} + 18241321144346224772t^{12} + 12827052414805691388t^{11} + 6870802410354954996t^{10} + 170529088759675779t^{9} + 267606967317991040t^{8} + 41228012708991203t^{7} + 2577758183175538t^{6} + 242895860908305t^{5} + 6395361463220t^{4} + 95295419031t^{3} + 2246964840t^{2} + 122562t + 21384 \right)D^{3} + t ^{2} \left( 634006187524957606464t^{20} + 11901572829066495316616t^{19} + 18722716110543803751744t^{18} + 4715313739585284206672t^{17} + 649074804437672362048t^{16} + 507286612170637880880t^{15} + 2639132050517385689964t^{14} + 489140865821356962228t^{13} + 1012457924976945962984t^{12} + 48586289580214524847t^{11} + 14089401530526478836t^{10} + 14188596781011811742t^{9} + 612069533674919018t^{8} + 463107826701470450t^{7} + 49313073787018546t^{6} + 3355030735851638t^{5} + 322585231891786t^{4} + 6962891697451t^{3} + 146149554602t^{2} + 3083881644t - 11232 \right)D^{2} + 2 t ^{3} \left( 344330946673037320752t^{19} + 6554825008081770798616t^{18} + 10347209690811534691696t^{17} + 2057443936012497898736t^{16} - 391105559358966631284t^{15} + 358578202230954083392t^{14} + 1858305615435126701788t^{13} + 308652697974461677592t^{12} + 491330336311475681884t^{11} + 30281211982295811389t^{10} + 3928241494672658668t^{9} + 7785268531331207008t^{8} + 393453052292017172t^{7} + 203192864398009998t^{6} + 15435868687321860t^{5} + 1234212349537584t^{4} + 113591302276176t^{3} + 2016286745517t^{2} + 57247821792t + 976906656 \right)D + 24 t ^{3} \left( 11711936961667936080t^{19} + 225045278823199825240t^{18} + 356117499368661970960t^{17} + 59334952391584157384t^{16} - 28935966218640727596t^{15} + 15084185333562440176t^{14} + 74091663719965943740t^{13} + 11468743189842147584t^{12} + 15232548170569825456t^{11} + 1133856377720054633t^{10} + 74094210089917972t^{9} + 270567945727222063t^{8} + 14243103264098384t^{7} + 5855115206922498t^{6} + 330891752595438t^{5} + 32003265714075t^{4} + 2737605212010t^{3} + 41637479679t^{2} + 1508925024t + 20785248 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 4)

|λ| = 6.376564 λ ≈ 6.376564; multiplicity 1; Jordan blocks 1 λ ≈ -0.7775937 - 5.939058i; multiplicity 1; Jordan blocks 1 λ ≈ -0.7775937 + 5.939058i; multiplicity 1; Jordan blocks 1 λ ≈ -4.579525 - 0.9642267i; multiplicity 1; Jordan blocks 1 λ ≈ -4.579525 + 0.9642267i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7634704 - 1.969653i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7634704 + 1.969653i; multiplicity 1; Jordan blocks 1 λ ≈ -0.09463345 - 0.9576831i; multiplicity 1; Jordan blocks 1 λ ≈ -0.09463345 + 0.9576831i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$6.37656429$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.376564$11
$-0.7775937 - 5.939058\mathrm{i}$11
$-0.7775937 + 5.939058\mathrm{i}$11
$-4.579525 - 0.9642267\mathrm{i}$11
$-4.579525 + 0.9642267\mathrm{i}$11
$0.7634704 - 1.969653\mathrm{i}$11
$0.7634704 + 1.969653\mathrm{i}$11
$-0.09463345 - 0.9576831\mathrm{i}$11
$-0.09463345 + 0.9576831\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\lambda^{9} + 3 \lambda^{8} + 3 \lambda^{7} - 80 \lambda^{6} - 1377 \lambda^{5} - 3445 \lambda^{4} - 1013 \lambda^{3} - 25081 \lambda^{2} - 3456 \lambda - 20707\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{9} + 3 \lambda^{8} + 3 \lambda^{7} - 80 \lambda^{6} - 1377 \lambda^{5} - 3445 \lambda^{4} - 1013 \lambda^{3} - 25081 \lambda^{2} - 3456 \lambda - 20707\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 3 & 4 & 4 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 4 \\ 0 & 3 & 4 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 1 & 0 & -1 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 3 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$