Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-433-90-a

·

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

  • $\pi_1\colon X\to Y$ maps birationally onto a fourfold of degree $8$ in $\mathrm{Gr}(3,5)$ with a single singular point; it contracts a plane with normal bundle $\mathcal O(-1)^{\oplus 2}$ to that point.

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

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 3, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
2 copies of E^(w2)
Bundle
2 copies of U_2^v ot U_3^v
ZL62 label
44.11
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 90
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 433
$\chi(X, \mathrm{T}_X)$ = 13

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 433
$\mathrm{c}_1^2\mathrm{c}_2$ = 202
$\mathrm{c}_2^2$ = 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
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 13 0 x0 - 13 x1
$\mathrm{HH}^{ 3 }$ = x1 + 69 0 0 x1 - 68 137
$\mathrm{HH}^{ 4 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 85t + 261t^{2} + 85t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(15)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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( 4t^{2} + 1 \right) \left( 24164t^{7} - 16200t^{6} + 4125t^{5} - 216t^{4} + 222t^{3} - 4t^{2} + t - 1 \right) \left( 163994044079424t^{12} - 155443967926992t^{11} + 353573768343192t^{10} + 18121300401732t^{9} - 129245093728674t^{8} + 70441912655968t^{7} - 13374467478642t^{6} + 1959821907738t^{5} - 227032384569t^{4} + 25554949395t^{3} - 1325123112t^{2} + 24333599t + 116910 \right)D^{6} + \left( 332871174815356929024t^{21} - 511196866564822930944t^{20} + 1054027945218367553664t^{19} - 506962424183529051648t^{18} - 89491724666086455840t^{17} + 293610646851487375264t^{16} - 216013548604975976904t^{15} + 106028195739865059312t^{14} - 36783051362097195882t^{13} + 9158440986864967200t^{12} - 1645528143600018258t^{11} + 263919264290466376t^{10} - 34383932357898599t^{9} + 2541031288116612t^{8} + 347629034425960t^{7} - 86449154989182t^{6} + 13796666909601t^{5} - 1564891089653t^{4} + 154113939254t^{3} - 6630291960t^{2} + 97451306t + 350730 \right)D^{5} + \left( 2773926456794641075200t^{21} - 4076756127630622010880t^{20} + 8756670416543092070400t^{19} - 3396129651157887426624t^{18} - 1849290392999640989280t^{17} + 2988730124484531895488t^{16} - 1780446086645094040080t^{15} + 726975672302888668844t^{14} - 216554675360699377770t^{13} + 48508944545207268408t^{12} - 8253298379326999442t^{11} + 1131222336413865414t^{10} - 144710831063192663t^{9} + 20113009216613477t^{8} - 2995657893124510t^{7} + 367535088509991t^{6} - 37160567431123t^{5} + 2768530182105t^{4} - 213890942839t^{3} + 7136335283t^{2} - 103717547t - 233820 \right)D^{4} + t \left( 11650491118537492515840t^{20} - 16517757126697034480640t^{19} + 36875644905320050496640t^{18} - 11571155142742205858304t^{17} - 11205650204195159821920t^{16} + 14006077532196855032768t^{15} - 7204023091399054163880t^{14} + 2562060150471586451808t^{13} - 671905914938238389430t^{12} + 138894786225929133576t^{11} - 24027115779106266630t^{10} + 3383188481569086284t^{9} - 403015474433046641t^{8} + 39691195610685378t^{7} - 2296449592912884t^{6} + 118238017257930t^{5} - 6631710551505t^{4} + 231666909854t^{3} + 1607185844t^{2} + 1722114t - 270 \right)D^{3} + 2 t ^{2} \left( 12871018759527134588928t^{19} - 17737142143970509098624t^{18} + 40936886456992341589056t^{17} - 10518101820638821217856t^{16} - 15149725166067241418112t^{15} + 16533477570747186966392t^{14} - 7651297018802471795820t^{13} + 2452978846440402574336t^{12} - 585961535124110568894t^{11} + 114119011253456085636t^{10} - 20205964250718840396t^{9} + 2981007926146476620t^{8} - 345137037052960713t^{7} + 31885628650443040t^{6} - 1539616589471661t^{5} + 44109577985124t^{4} - 1631768454168t^{3} + 117578122100t^{2} + 836227626t + 277120 \right)D^{2} + 72 t ^{3} \left( 388349703951249750528t^{18} - 523872647496838991808t^{17} + 1241315703379522592928t^{16} - 267281079565348601568t^{15} - 516114026297363170160t^{14} + 520246266248000506612t^{13} - 223517571960627119430t^{12} + 66492612827519665520t^{11} - 14948568859307411900t^{10} + 2802827284743705874t^{9} - 502904307623692694t^{8} + 77280012422356688t^{7} - 8789551666454112t^{6} + 784043043579814t^{5} - 35061934040859t^{4} + 597411689331t^{3} + 7231714812t^{2} + 1740833685t + 13468032 \right)D + 360 t ^{3} \left( 31702016649081612288t^{18} - 42141924420117573888t^{17} + 101730676984036597632t^{16} - 19054840520939835648t^{15} - 45307769013387176832t^{14} + 43590733757764156592t^{13} - 17835157444957254024t^{12} + 5045896781179139464t^{11} - 1094636714591022852t^{10} + 201011228748603792t^{9} - 36194112395504274t^{8} + 5705964867392764t^{7} - 639957677045676t^{6} + 55994045415176t^{5} - 2464469003493t^{4} + 35174161302t^{3} + 1958747664t^{2} + 79147584t + 631314 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 6.356331 λ ≈ 6.356331; multiplicity 1; Jordan blocks 1 λ ≈ -4.04181 - 4.418026i; multiplicity 1; Jordan blocks 1 λ ≈ -4.04181 + 4.418026i; multiplicity 1; Jordan blocks 1 λ ≈ -0.4441956 - 4.754189i; multiplicity 1; Jordan blocks 1 λ ≈ -0.4441956 + 4.754189i; multiplicity 1; Jordan blocks 1 λ ≈ 1.80784 - 1.175599i; multiplicity 1; Jordan blocks 1 λ ≈ 1.80784 + 1.175599i; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 2i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$6.35633099$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.356331$11
$-4.04181 - 4.418026\mathrm{i}$11
$-4.04181 + 4.418026\mathrm{i}$11
$-0.4441956 - 4.754189\mathrm{i}$11
$-0.4441956 + 4.754189\mathrm{i}$11
$1.80784 - 1.175599\mathrm{i}$11
$1.80784 + 1.175599\mathrm{i}$11
$0 - 2\mathrm{i}$11
$0 + 2\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 4\right) \left(\lambda^{7} - \lambda^{6} + 4 \lambda^{5} - 222 \lambda^{4} + 216 \lambda^{3} - 4125 \lambda^{2} + 16200 \lambda - 24164\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{2} + 4\right) \left(\lambda^{7} - \lambda^{6} + 4 \lambda^{5} - 222 \lambda^{4} + 216 \lambda^{3} - 4125 \lambda^{2} + 16200 \lambda - 24164\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 6 & 6 & 3 & 0 & 0 & 10 \\ 1 & 0 & 0 & 0 & 0 & 4 & -6 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & -2 & 6 & 3 & 0 \\ 0 & 2 & 2 & 0 & -\frac{3}{2} & 0 & -2 & 0 & -3 \\ 0 & 1 & 0 & 0 & 1 & 0 & 4 & 0 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & 5 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$