Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 6-172-40-a

·

Successive blow-ups of a smooth quadric fourfold along five incident planes.

  • $\pi_1\colon X\to Y\subset\mathrm{Gr}(2,5)$ maps birationally onto a determinantal fourfold of degree $12$ with ten singular points, each locally the cone over $\mathbb P^1\times\mathbb P^2$; it contracts ten disjoint planes with normal bundle $\mathcal O_{\mathbb P^2}(-1)^{\oplus 2}$ to these points.

  • $\pi_2\colon X\to Q$ is the blow-up of the smooth quadric fourfold $Q=\mathrm{Gr}(2,4)$ (1-512-105-a) along five planes $\Pi_1,\ldots,\Pi_5$ of the same family, meeting pairwise in the ten points $p_{ij}$, performed successively along strict transforms; the fibers are $\mathbb P^1$ over a general point of each $\Pi_i$ and $\mathbb P^1\times\mathbb P^1$ over the points $p_{ij}$.

Varieties in this family can be blown down to

Defining data
Ambient
Gr(2, 4) × Gr(2, 5)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w1 + w6)
Bundle
U_1^v box Q_2^v(1)
ZL62 label
3141.w4
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;5) \\ \det\mathcal{E} &\cong \mathcal{O}(3;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 40
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 172
$\chi(X, \mathrm{T}_X)$ = 0

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 172
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 123
$\mathrm{c}_1\mathrm{c}_3$ = 58
$\mathrm{c}_4$ = 31
Hodge diamond
1
00
0 6 0
0 0 0 0
0 0 17 0 0
0 0 0 0
0 6 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 0 x0 x1
$\mathrm{HH}^{ 3 }$ = x1 + 10 0 0 x1 - 10 20
$\mathrm{HH}^{ 4 }$ = 40 0 0 0 0 40
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 35t + 100t^{2} + 35t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
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( t + 1 \right) ^{3} \left( 28t^{2} + 2t + 1 \right) \left( 13666t^{5} + 2399t^{4} + 650t^{3} + 44t^{2} + t - 1 \right) \left( 1212857425006080t^{15} + 11174699927762688t^{14} + 24570868978580992t^{13} + 23078795737543168t^{12} + 10151876545370248t^{11} + 2912832930454640t^{10} + 1169170075991300t^{9} + 435801114191404t^{8} + 83459559133730t^{7} + 8006930821424t^{6} + 803133132695t^{5} + 90368550653t^{4} + 7110090311t^{3} + 259129955t^{2} + 1295704t + 47128 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 9746046827238256496640t^{23} + 101125094790265974263808t^{22} + 287946972974935862484992t^{21} + 392661693261562122222080t^{20} + 300970351706294087669696t^{19} + 148628288989730244411264t^{18} + 59511074679183995533856t^{17} + 23610181862319664035000t^{16} + 8406145064677560855880t^{15} + 2335284687846347904212t^{14} + 505970370249298183844t^{13} + 87002439135908005166t^{12} + 11827399509893143730t^{11} + 1410627741935655247t^{10} + 162114365460487709t^{9} + 18842616166317995t^{8} + 1873672092646698t^{7} + 126380863806376t^{6} + 7552993727870t^{5} + 543094858652t^{4} + 34472605785t^{3} + 1030678476t^{2} + 3415832t + 94256 \right)D^{5} + \left( t + 1 \right) \left( 81217056893652137472000t^{24} + 918436329385536414351360t^{23} + 3061827734420119716198400t^{22} + 5033892876856190491127296t^{21} + 4774012890171382997921088t^{20} + 2870456080786359143708608t^{19} + 1260537470205771458316512t^{18} + 506627957072735652110344t^{17} + 201660673028168062968064t^{16} + 66844328627177662466012t^{15} + 16338689709671190466716t^{14} + 3001309269637266222646t^{13} + 441856141823964827620t^{12} + 54016661596642868669t^{11} + 6083669364115530397t^{10} + 625190782356522560t^{9} + 48349423286137509t^{8} + 2503050947458613t^{7} + 110034904267552t^{6} + 2423555535004t^{5} - 326546984935t^{4} - 32474936720t^{3} - 1009384451t^{2} - 4745048t - 47128 \right)D^{4} + t \left( 341111638953338977382400t^{24} + 4180427714863495739105280t^{23} + 16079641220228375539220480t^{22} + 31286645255568402551901696t^{21} + 35954666932968245610044224t^{20} + 26430930455056093994043648t^{19} + 13573937209779925792226080t^{18} + 5735990413520528153604648t^{17} + 2372288624752207764019272t^{16} + 911606341375382778953380t^{15} + 272547331069371618097900t^{14} + 58568000627407948642358t^{13} + 9426613217166144265754t^{12} + 1269887274061864315775t^{11} + 153654061937992112881t^{10} + 17290689765563983198t^{9} + 1739946262808785509t^{8} + 139952010554683803t^{7} + 8374595631951856t^{6} + 349037702264080t^{5} + 9692722664139t^{4} + 138254488700t^{3} + 3533698321t^{2} + 28515130t + 32880 \right)D^{3} + 2 t ^{2} \left( 376847143986545917870080t^{23} + 4603064522692697332282368t^{22} + 17210544332755683316321280t^{21} + 32263871319496401433381888t^{20} + 35425543785115877477846272t^{19} + 24754832119889542900478336t^{18} + 12212918818548235605689312t^{17} + 5170240820233820994415888t^{16} + 2209542331212858008786864t^{15} + 850433661344925101300200t^{14} + 244162351895524588963600t^{13} + 48982773410793121033376t^{12} + 7292621830736216171268t^{11} + 944843973249894453794t^{10} + 116109873564386261786t^{9} + 12866805787341337919t^{8} + 1206975017474252278t^{7} + 89360777658073145t^{6} + 4977464411513146t^{5} + 194384290203031t^{4} + 5611035419436t^{3} + 101595297731t^{2} + 2223591214t + 8428860 \right)D^{2} + 4 t ^{2} \left( 204666983372003386429440t^{23} + 2493845770672225483250688t^{22} + 9122673643931253135193088t^{21} + 16606321926784315082251776t^{20} + 17569555642063397471264448t^{19} + 11776694895779182010480256t^{18} + 5636202338961329039814144t^{17} + 2407193063830052181035560t^{16} + 1056744696602550766377720t^{15} + 404009565245129042963252t^{14} + 111520202358258465250052t^{13} + 21148749540863638766862t^{12} + 2981130228438308355494t^{11} + 379511573921110450177t^{10} + 47415905041167039557t^{9} + 5119881472254259869t^{8} + 448754043247636069t^{7} + 31242494421403656t^{6} + 1663516053744782t^{5} + 61380883819748t^{4} + 1792182336514t^{3} + 35309596136t^{2} + 619779232t + 942560 \right)D + 48 t ^{3} \left( 6961462019455897497600t^{22} + 84684638864012801080320t^{21} + 305080409965156385469440t^{20} + 543779962950252650672640t^{19} + 559802288096975805703488t^{18} + 363697523905565465236480t^{17} + 170347340768297683796928t^{16} + 73503269175894599555608t^{15} + 32893125565224443988272t^{14} + 12458071023339116656268t^{13} + 3326628937211042638360t^{12} + 603987610676506633732t^{11} + 82077692151967178820t^{10} + 10388682841733687826t^{9} + 1312652977268361316t^{8} + 137844160282714272t^{7} + 11421070841839486t^{6} + 762208338613349t^{5} + 39510426614249t^{4} + 1389581825953t^{3} + 40437238853t^{2} + 825102468t + 11452104 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 5)

|λ| = 11.71309 λ ≈ 11.71309; multiplicity 1; Jordan blocks 1 λ ≈ -4.782046 - 4.590843i; multiplicity 1; Jordan blocks 1 λ ≈ -4.782046 + 4.590843i; multiplicity 1; Jordan blocks 1 λ ≈ -1 - 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ -1 + 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5744993 - 5.120588i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5744993 + 5.120588i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
10
Spectral radius
$11.7130908$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.71309$11
$-4.782046 - 4.590843\mathrm{i}$11
$-4.782046 + 4.590843\mathrm{i}$11
$-1 - 5.196152\mathrm{i}$11
$-1 + 5.196152\mathrm{i}$11
$-0.5744993 - 5.120588\mathrm{i}$11
$-0.5744993 + 5.120588\mathrm{i}$11
$-1$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{2} + 2 \lambda + 28\right) \left(\lambda^{5} - \lambda^{4} - 44 \lambda^{3} - 650 \lambda^{2} - 2399 \lambda - 13666\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{2} + 2 \lambda + 28\right) \left(\lambda^{5} - \lambda^{4} - 44 \lambda^{3} - 650 \lambda^{2} - 2399 \lambda - 13666\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 24 & 36 & 3 & 18 & 132 & 60 & 150 \\ 1 & 0 & 3 & -4 & -4 & 0 & 0 & 9 & 15 & -12 \\ 1 & 0 & -1 & 8 & 8 & 0 & 0 & 27 & 15 & 48 \\ 0 & 1 & 0 & 1 & 3 & 0 & 0 & 6 & 2 & 18 \\ 0 & 1 & 2 & 0 & -1 & 0 & 2 & 12 & 4 & 9 \\ 0 & 1 & -3 & -6 & 3 & 0 & -2 & -48 & -16 & -36 \\ 0 & 0 & 2 & 3 & 0 & 0 & -2 & 6 & 2 & 9 \\ 0 & 0 & 0 & \frac{4}{3} & \frac{4}{3} & \frac{1}{2} & 0 & 0 & \frac{5}{3} & 0 \\ 0 & 0 & 0 & 0 & 3 & \frac{1}{2} & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & \frac{10}{3} & 0\end{pmatrix}\notag\end{equation}$$