Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-512-105-b

·

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

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

  • $\pi_2\colon X\to\mathbb P^1$ is the $\mathbb P^3$-bundle $\mathbb P_{\mathbb P^1}(\mathcal O^{\oplus 3}\oplus\mathcal O(-1))$.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

Defining data
Ambient
ℙ1P1 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2)
Bundle
O_1(1) box O_2(1)
ZL62 label
140.Z
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 105
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 512
$\chi(X, \mathrm{T}_X)$ = 18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 512
$\mathrm{c}_1^2\mathrm{c}_2$ = 224
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 56
$\mathrm{c}_4$ = 8
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 2 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
$\mathrm{HH}^{ 2 }$ = 90 0 0 90
$\mathrm{HH}^{ 3 }$ = 170 0 0 0 170
$\mathrm{HH}^{ 4 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 100t + 310t^{2} + 100t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(2)
Coates et al. (2020) Obro(4,115)
Coates–Kasprzyk–Prince (2015) CKP(2)

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( 62411t^{8} - 10000t^{7} - 11200t^{6} - 4864t^{5} - 512t^{4} + t^{3} + 3t^{2} + 3t + 1 \right) \left( 7082637345280t^{9} + 18271445739520t^{8} + 16227865660655t^{7} + 4973392247037t^{6} + 45001716325t^{5} + 25307054432t^{4} + 4382457056t^{3} + 2237120t^{2} + 537280t + 51200 \right)D^{6} + \left( 9282724066481671680t^{17} + 23883414052384419840t^{16} + 18974246215597833715t^{15} + 1191158504807665688t^{14} - 4557599499832620225t^{13} - 1783995371151381328t^{12} - 355225843573097184t^{11} - 29532201543506227t^{10} - 3298413486566809t^{9} - 875763190805000t^{8} - 230440430457044t^{7} - 40080421051157t^{6} - 435453128483t^{5} - 165325018400t^{4} - 21903714080t^{3} - 6415040t^{2} - 1304640t - 102400 \right)D^{5} + \left( 77356033887347264000t^{17} + 208519415062272716800t^{16} + 181111033894811581835t^{15} + 30575289799950507937t^{14} - 26347192888579551665t^{13} - 8940024765923760800t^{12} - 1500996674869508464t^{11} - 116838442228542562t^{10} - 16456976047862471t^{9} - 1798830276545597t^{8} + 238039558923533t^{7} + 78345107377298t^{6} + 74866342813t^{5} + 125784179776t^{4} + 22058281744t^{3} + 6338560t^{2} + 1028160t + 51200 \right)D^{4} + t \left( 324895342326858508800t^{16} + 907101726526473830400t^{15} + 840549399763937748625t^{14} + 198583412255635017172t^{13} - 79183707617782619975t^{12} - 22851862198234851200t^{11} - 3114590774443471744t^{10} - 265684578314152192t^{9} - 51404267272555371t^{8} - 6596601418421824t^{7} - 260887669978564t^{6} - 1623328764279t^{5} - 152583579669t^{4} - 4144422752t^{3} + 31680t^{2} - 20800t - 4800 \right)D^{3} + t ^{2} \left( 717863994474582609920t^{15} + 2057208886511111649280t^{14} + 1997224754863616215520t^{13} + 561705792403181020140t^{12} - 129909325009490980630t^{11} - 31158785770714417312t^{10} - 3202132167883326032t^{9} - 377870973958121728t^{8} - 84102120633681735t^{7} - 9686989176007656t^{6} - 301148670430858t^{5} - 3211816943906t^{4} - 313732417482t^{3} - 10753472608t^{2} + 60400t + 5120 \right)D^{2} + 4 t ^{4} \left( 194937205396115105280t^{13} + 569174818058440995840t^{12} + 570897513412816972615t^{11} + 177751384181422542756t^{10} - 27399605235330954790t^{9} - 5324646549131726088t^{8} - 346214553092470960t^{7} - 78633599178947520t^{6} - 17556053664189888t^{5} - 1816363114676224t^{4} - 41759142866944t^{3} - 774972219392t^{2} - 72342310912t - 2229534720 \right)D + 240 t ^{4} \left( 1326103438068810240t^{13} + 3920360826719339520t^{12} + 4016779806434545301t^{11} + 1327458332068367148t^{10} - 153175142177994050t^{9} - 23719918422351396t^{8} - 488936065009676t^{7} - 464932312784736t^{6} - 96830358364800t^{5} - 9066308986112t^{4} - 143462618624t^{3} - 4824690688t^{2} - 431679488t - 11796480 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ4P4

|λ| = 5.692888 λ ≈ 5.692888; multiplicity 1; Jordan blocks 1 λ ≈ 1.094495 - 5.369507i; multiplicity 1; Jordan blocks 1 λ ≈ 1.094495 + 5.369507i; multiplicity 1; Jordan blocks 1 λ ≈ -4.364181 - 1.920759i; multiplicity 1; Jordan blocks 1 λ ≈ -4.364181 + 1.920759i; multiplicity 1; Jordan blocks 1 λ ≈ -1.846447 - 2.649863i; multiplicity 1; Jordan blocks 1 λ ≈ -1.846447 + 2.649863i; multiplicity 1; Jordan blocks 1 λ ≈ 1.539379; multiplicity 1; Jordan blocks 1
Sector dimension
8
Spectral radius
$5.69288796$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$5.692888$11
$1.094495 - 5.369507\mathrm{i}$11
$1.094495 + 5.369507\mathrm{i}$11
$-4.364181 - 1.920759\mathrm{i}$11
$-4.364181 + 1.920759\mathrm{i}$11
$-1.846447 - 2.649863\mathrm{i}$11
$-1.846447 + 2.649863\mathrm{i}$11
$1.539379$11

Characteristic polynomial

$$\begin{equation}\lambda^{8} + 3 \lambda^{7} + 3 \lambda^{6} + \lambda^{5} - 512 \lambda^{4} - 4864 \lambda^{3} - 11200 \lambda^{2} - 10000 \lambda + 62411\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{8} + 3 \lambda^{7} + 3 \lambda^{6} + \lambda^{5} - 512 \lambda^{4} - 4864 \lambda^{3} - 11200 \lambda^{2} - 10000 \lambda + 62411\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 4 & 5 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 4 \\ 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 4 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 5 & 0\end{pmatrix}\notag\end{equation}$$