Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-544-111-a

·

Blow-up of $\mathbb P^4$ at a point.

  • $\pi_1\colon X\to\mathbb P^3$ is the $\mathbb P^1$-bundle $\mathbb P_{\mathbb P^3}(\mathcal O\oplus\mathcal O(-1))$, resolving the projection of $\mathbb P^4$ (1-625-126-a) from a point.

  • $\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) at a point.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w4)
Bundle
wedge^2 Q_1^v ot U_2^v ot U_3^v
ZL62 label
42.3
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,4) \\ \det\mathcal{E} &\cong \mathcal{O}(0,1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 111
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 544
$\chi(X, \mathrm{T}_X)$ = 20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 544
$\mathrm{c}_1^2\mathrm{c}_2$ = 232
$\mathrm{c}_2^2$ = 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
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + 80 0 x0 - 20 100
$\mathrm{HH}^{ 3 }$ = 184 0 0 0 184
$\mathrm{HH}^{ 4 }$ = 111 0 0 0 0 111
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 106t + 330t^{2} + 106t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(35)
Coates et al. (2020) Obro(4,121)
Coates–Kasprzyk–Prince (2015) CKP(47)

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( 238531217250t^{7} - 5953418050560t^{6} + 59383353825t^{5} - 120371608560t^{4} - 97793372160t^{3} - 2900790608t^{2} + 42301440t - 22282240 \right) \left( 84119t^{8} + 18000t^{7} + 256t^{6} + 3800t^{5} - 96t^{4} - 27t^{3} + 16t^{2} - 1 \right)D^{6} + \left( 421365156740907750t^{15} - 10944512053412746080t^{14} - 1813222444661112975t^{13} - 235883119826114400t^{12} - 475107144726648000t^{11} - 37898285875013102t^{10} - 6697050143765760t^{9} - 4601183983237169t^{8} - 64574916822170t^{7} - 41250219269120t^{6} - 1959746204279t^{5} - 1038123999616t^{4} - 677710172160t^{3} - 18830807008t^{2} + 211507200t - 89128960 \right)D^{5} + \left( 3511376306174231250t^{15} - 94657399249353261600t^{14} - 12563987053895034375t^{13} - 2135782876602266160t^{12} - 3213782353671806400t^{11} - 357775662188176680t^{10} - 34493369743845840t^{9} - 24603588025601910t^{8} - 700118453305950t^{7} + 245712398240384t^{6} - 2636505845995t^{5} + 1303685581968t^{4} + 1170332559360t^{3} + 24401344464t^{2} - 42301440t + 89128960 \right)D^{4} + t ^{2} \left( 14747780485931771250t^{13} - 408956412901298944800t^{12} - 43517229038298589125t^{11} - 9430072337970125760t^{10} - 12750280554147316800t^{9} - 1576547563731307960t^{8} - 82021107356195040t^{7} - 68157032001802240t^{6} - 2507617480239310t^{5} + 129103664622464t^{4} - 14671537746085t^{3} + 517413868224t^{2} - 20692869120t - 2296381440 \right)D^{3} + 8 t ^{2} \left( 4073196515162108250t^{13} - 115357809455582215920t^{12} - 9932506866990017100t^{11} - 2687922498955849320t^{10} - 3614366977743148020t^{9} - 443500763317037896t^{8} - 11718567976879320t^{7} - 12592820048895904t^{6} - 593669238441925t^{5} + 18502704597952t^{4} - 5717669363712t^{3} + 68052460640t^{2} - 4060938240t - 89128960 \right)D^{2} + 12 t ^{3} \left( 2949556097186354250t^{12} - 84812773200642619360t^{11} - 6036841558001583825t^{10} - 1986123387641256280t^{9} - 2754312306268307400t^{8} - 322877556670173444t^{7} - 3401016269294340t^{6} - 6325098491854208t^{5} - 374193946345060t^{4} + 6254098194112t^{3} - 4441099051008t^{2} + 16712742976t - 1974067200 \right)D + 240 t ^{3} \left( 60195022391558250t^{12} - 1748490943572198240t^{11} - 106860012485772825t^{10} - 41044162274480220t^{9} - 58814139764164380t^{8} - 6591459991760916t^{7} - 4600991276220t^{6} - 95238488929792t^{5} - 6861385235823t^{4} + 51053888096t^{3} - 88727261184t^{2} + 17523872t - 28200960 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5)

|λ| = 5.780046 λ ≈ 5.780046; multiplicity 1; Jordan blocks 1 λ ≈ -4.807446 - 2.601243i; multiplicity 1; Jordan blocks 1 λ ≈ -4.807446 + 2.601243i; multiplicity 1; Jordan blocks 1 λ ≈ 2.249567 - 3.96711i; multiplicity 1; Jordan blocks 1 λ ≈ 2.249567 + 3.96711i; multiplicity 1; Jordan blocks 1 λ ≈ 0.8075677 - 3.101963i; multiplicity 1; Jordan blocks 1 λ ≈ 0.8075677 + 3.101963i; multiplicity 1; Jordan blocks 1 λ ≈ -2.279424; multiplicity 1; Jordan blocks 1
Sector dimension
8
Spectral radius
$5.78004612$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$5.780046$11
$-4.807446 - 2.601243\mathrm{i}$11
$-4.807446 + 2.601243\mathrm{i}$11
$2.249567 - 3.96711\mathrm{i}$11
$2.249567 + 3.96711\mathrm{i}$11
$0.8075677 - 3.101963\mathrm{i}$11
$0.8075677 + 3.101963\mathrm{i}$11
$-2.279424$11

Characteristic polynomial

$$\begin{equation}\lambda^{8} - 16 \lambda^{6} + 27 \lambda^{5} + 96 \lambda^{4} - 3800 \lambda^{3} - 256 \lambda^{2} - 18000 \lambda - 84119\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{8} - 16 \lambda^{6} + 27 \lambda^{5} + 96 \lambda^{4} - 3800 \lambda^{3} - 256 \lambda^{2} - 18000 \lambda - 84119\notag\end{equation}$$

Quantum multiplication matrix

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