Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-115-30-a

·

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

  • $\pi_1\colon X\to W$ is the blow-up of a Gushel–Mukai fourfold $W$ (1-160-39-a) along a smooth quadric surface contained in it.

  • $\pi_2\colon X\to\mathbb P^4$ resolves the projection of $W\subset\mathbb P^8$ (1-160-39-a) from the $\mathbb P^3$ spanned by the quadric surface onto $\mathbb P^4$ (1-625-126-a); its center is a surface of degree $8$ and sectional genus $6$ with one improper double point, whose normalization has the invariants of a K3 surface of degree $10$ blown up in two points. The fibers are lines over the rest of the surface and a smooth quadric surface over the double point.

This is a Fano fourfold of K3 type, appearing as case GM-23 in Bernardara–Fatighenti–Manivel–Tanturri.

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
E^(w2) + E^(w3) + E^(w1 + w3)
Bundle
U_2^v ot U_3^v + O(0,1) + O(1,1)
ZL62 label
44.21T
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 115
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 115
$\mathrm{c}_1^2\mathrm{c}_2$ = 118
$\mathrm{c}_2^2$ = 125
$\mathrm{c}_1\mathrm{c}_3$ = 20
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 24 1 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 + x2 + 23 0 x0 + 23 x2
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 15
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 23 0 0 x1 - x2 - 7 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 63t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( 2t + 1 \right) \left( t + 1 \right) ^{3} \left( 9257t^{6} + 15969t^{5} + 10304t^{4} + 2955t^{3} + 201t^{2} - 4t - 1 \right) \left( 204498032904t^{14} + 3477843275544t^{13} + 12586425439884t^{12} + 20653921438824t^{11} + 17722775828442t^{10} + 6736004903580t^{9} - 1507211406945t^{8} - 3127772536473t^{7} - 1669441927163t^{6} - 472785461484t^{5} - 78232216030t^{4} - 8953545807t^{3} - 842357669t^{2} - 6930836t - 43815 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 79507608204877776t^{22} + 1605199347762656184t^{21} + 8940120087023124240t^{20} + 25421206179307884036t^{19} + 44249511773723918400t^{18} + 50401638910052153334t^{17} + 37229414907106040880t^{16} + 14889585304462645227t^{15} - 1641911634292904179t^{14} - 6968495799142138400t^{13} - 5371946520844201913t^{12} - 2524609603028604525t^{11} - 825886503078166264t^{10} - 196246192691822561t^{9} - 34822913396109680t^{8} - 4752884796411327t^{7} - 497801047597673t^{6} - 35428638878268t^{5} - 1332982142961t^{4} - 48426061245t^{3} - 3334106624t^{2} - 19960023t - 87630 \right)D^{5} + \left( t + 1 \right) \left( 662563401707314800t^{23} + 14016670735135040280t^{22} + 83325728036349835128t^{21} + 254329799763660953340t^{20} + 477559572410018300544t^{19} + 590970574618208587242t^{18} + 481271336975588495136t^{17} + 223987078847795092497t^{16} - 1756786024936818354t^{15} - 95694761447111740509t^{14} - 86170313884991061898t^{13} - 45994196553705126740t^{12} - 17125636891164912659t^{11} - 4674753783316253794t^{10} - 961738024558242867t^{9} - 153047641853121413t^{8} - 18986999749517763t^{7} - 1708167083895038t^{6} - 88280614391245t^{5} - 1406805207067t^{4} + 14891339344t^{3} + 3152382531t^{2} + 18778858t + 43815 \right)D^{4} + t \left( 2782766287170722160t^{23} + 61578355511576340360t^{22} + 393222559546326171120t^{21} + 1299984963189893432628t^{20} + 2663353974610064457048t^{19} + 3633297441175347804462t^{18} + 3329812806115171985376t^{17} + 1867858227610201440819t^{16} + 270203283764643737797t^{15} - 601579681863528964433t^{14} - 690575308098909050282t^{13} - 430917907139081900319t^{12} - 185423338359158570661t^{11} - 58592819026671342080t^{10} - 14002369397257767920t^{9} - 2594615302035000708t^{8} - 380447887001653007t^{7} - 43315975525376747t^{6} - 3425143759156372t^{5} - 147723918643011t^{4} - 2423140212649t^{3} - 30466055114t^{2} - 198192514t - 53721 \right)D^{3} + 2 t ^{2} \left( 3074294183921940672t^{22} + 67966512061783503168t^{21} + 420896258070226291680t^{20} + 1338113531771416559736t^{19} + 2617444472393125266120t^{18} + 3372752096150619398658t^{17} + 2851527393983760454482t^{16} + 1353206772719457481989t^{15} - 75283701656945279765t^{14} - 722161835064689537974t^{13} - 672257076794530766503t^{12} - 379500005490864500951t^{11} - 151121667074091424136t^{10} - 44579836946893315489t^{9} - 10028110743346416466t^{8} - 1768992554295310591t^{7} - 248391256553067079t^{6} - 26578665452200008t^{5} - 1877274667229305t^{4} - 67571060834155t^{3} - 1015014012344t^{2} - 14606103427t - 75672300 \right)D^{2} + 24 t ^{2} \left( 278276628717072216t^{22} + 6147968220814246596t^{21} + 37186660281561443784t^{20} + 114709406677865065062t^{19} + 216394455039903854304t^{18} + 266343462131574664083t^{17} + 210196650536757639570t^{16} + 83959174132118182038t^{15} - 24600937558310766873t^{14} - 65635045328533013883t^{13} - 54171401296138166391t^{12} - 28494586407187925562t^{11} - 10698525392324340216t^{10} - 2991823039322530748t^{9} - 642369866998385954t^{8} - 109103925041396109t^{7} - 14747182606946759t^{6} - 1482296121603343t^{5} - 93602597245965t^{4} - 2856901331165t^{3} - 43178080808t^{2} - 585425181t - 2395220 \right)D + 72 t ^{2} \left( 37860765811846560t^{22} + 836074691642622960t^{21} + 4974777021297039456t^{20} + 15022636373624885640t^{19} + 27611942359137680472t^{18} + 32853260921630099442t^{17} + 24564490483411782366t^{16} + 8325948593138398398t^{15} - 4633172893743921244t^{14} - 8789443952701058765t^{13} - 6735284134791921611t^{12} - 3372373639412005081t^{11} - 1212544085641257730t^{10} - 325760509603670536t^{9} - 67545300692476600t^{8} - 11145810579279743t^{7} - 1458033055701874t^{6} - 138333001084378t^{5} - 7883906039111t^{4} - 209003886454t^{3} - 3336734484t^{2} - 38614223t - 116840 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 18.07142 λ ≈ 18.07142; multiplicity 1; Jordan blocks 1 λ ≈ -8.920185 - 5.964704i; multiplicity 1; Jordan blocks 1 λ ≈ -8.920185 + 5.964704i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.464885 - 1.128158i; multiplicity 1; Jordan blocks 1 λ ≈ -1.464885 + 1.128158i; multiplicity 1; Jordan blocks 1 λ ≈ -1.301283; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 3
Sector dimension
10
Spectral radius
$18.0714235$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.07142$11
$-8.920185 - 5.964704\mathrm{i}$11
$-8.920185 + 5.964704\mathrm{i}$11
$-2$11
$-1.464885 - 1.128158\mathrm{i}$11
$-1.464885 + 1.128158\mathrm{i}$11
$-1.301283$11
$-1$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda + 2\right) \left(\lambda^{6} + 4 \lambda^{5} - 201 \lambda^{4} - 2955 \lambda^{3} - 10304 \lambda^{2} - 15969 \lambda - 9257\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda + 2\right) \left(\lambda^{6} + 4 \lambda^{5} - 201 \lambda^{4} - 2955 \lambda^{3} - 10304 \lambda^{2} - 15969 \lambda - 9257\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 90 & 45 & 78 & 48 & 88 & 160 & 290 \\ 1 & -1 & 4 & 4 & 8 & 0 & 0 & 36 & 63 & 32 \\ 1 & 1 & -1 & 28 & 18 & 20 & 12 & 24 & 39 & 56 \\ 0 & 1 & 1 & -2 & -2 & 1 & 1 & 0 & 2 & 12 \\ 0 & 1 & 0 & 4 & 0 & 4 & 2 & 0 & 0 & 12 \\ 0 & 0 & 1 & 1 & 2 & -1 & 0 & 8 & 14 & 6 \\ 0 & 0 & 1 & 1 & 2 & -1 & -2 & 4 & 6 & 0 \\ 0 & 0 & 0 & -3 & -8 & 4 & 2 & -9 & -10 & 12 \\ 0 & 0 & 0 & 6 & 7 & 1 & 1 & 6 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$