Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-225-51-b

·

Blow-up of $\mathbb P^4$ along an elliptic normal quintic curve.

  • $\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along an elliptic quintic scroll.

  • $\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along an elliptic normal quintic curve. Together with the previous item it resolves the quadro-cubic Cremona transformation of $\mathbb P^4$.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

Defining data
Ambient
ℙ4P4 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 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_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w1 + w6)
Bundle
O_1(1) box Q_2^v(1)
ZL62 label
4040.w9
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;5) \\ \det\mathcal{E} &\cong \mathcal{O}(4;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 225
$\chi(X, \mathrm{T}_X)$ = -1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 225
$\mathrm{c}_1^2\mathrm{c}_2$ = 150
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 50
$\mathrm{c}_4$ = 5
Hodge diamond
1
00
0 2 0
0 1 1 0
0 0 3 0 0
0 1 1 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 + 1 0 x0 + 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 48 0 0 x1 - 1 49
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 46t + 131t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(32)
Coates et al. (2020) Str(1)
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( 10t - 1 \right) \left( 5t^{2} + 2t + 1 \right) ^{2} \left( 625t^{4} + 375t^{3} + 100t^{2} + 10t + 1 \right) \left( 29819718750t^{11} + 36658575000t^{10} + 24145921875t^{9} + 9044458125t^{8} + 2043090375t^{7} + 2011368320t^{6} + 1104396003t^{5} + 226346810t^{4} + 16354171t^{3} + 1159055t^{2} + 5912t + 144 \right)D^{6} + \left( 5t^{2} + 2t + 1 \right) \left( 19569190429687500t^{18} + 38342083886718750t^{17} + 38918359570312500t^{16} + 25241074716796875t^{15} + 11405854599609375t^{14} + 5135275034843750t^{13} + 2815130206593750t^{12} + 1405531502640625t^{11} + 507085845031250t^{10} + 122898022256250t^{9} + 19497098774400t^{8} + 2018569885535t^{7} + 145683491035t^{6} + 12741730706t^{5} + 1277326426t^{4} + 75391365t^{3} + 4580284t^{2} + 16296t + 288 \right)D^{5} + \left( 815382934570312500t^{20} + 1812827424316406250t^{19} + 2128777022460937500t^{18} + 1636002187646484375t^{17} + 901037784316406250t^{16} + 439066092339843750t^{15} + 228053963911328125t^{14} + 121071511300859375t^{13} + 54519170744531250t^{12} + 18577817862018750t^{11} + 4596200626031875t^{10} + 810288373047750t^{9} + 100051421231650t^{8} + 8890348918465t^{7} + 542797633601t^{6} + 10070744532t^{5} - 1517860010t^{4} - 72060359t^{3} - 4360141t^{2} - 16224t - 144 \right)D^{4} + t \left( 3424608325195312500t^{19} + 7247863674316406250t^{18} + 8038122571289062500t^{17} + 5773314697880859375t^{16} + 2938326951451171875t^{15} + 1431586199687890625t^{14} + 812060525558515625t^{13} + 445616163717734375t^{12} + 191791825529925000t^{11} + 59755243305281875t^{10} + 13198255383632750t^{9} + 2069053849706825t^{8} + 228956040788075t^{7} + 18666085190075t^{6} + 1109720227893t^{5} + 37613988121t^{4} + 382688974t^{3} + 6733951t^{2} + 27576t + 80 \right)D^{3} + 2 t ^{2} \left( 3783376816406250000t^{18} + 7697796808593750000t^{17} + 8160913334765625000t^{16} + 5547958723417968750t^{15} + 2643861147928125000t^{14} + 1317771759633125000t^{13} + 812580818401312500t^{12} + 449782736128921875t^{11} + 183962592668534375t^{10} + 52951378954184625t^{9} + 10611823106044575t^{8} + 1507158279894240t^{7} + 153041366210875t^{6} + 11654737926056t^{5} + 630446230713t^{4} + 19118459705t^{3} + 264351306t^{2} + 5114825t + 9180 \right)D^{2} + 4 t ^{2} \left( 2054764995117187500t^{18} + 4057572687011718750t^{17} + 4158007314257812500t^{16} + 2708461381904296875t^{15} + 1224557642030859375t^{14} + 631001732382421875t^{13} + 415991146092328125t^{12} + 229412642190937500t^{11} + 89704824483590625t^{10} + 24242002014041625t^{9} + 4494713688811250t^{8} + 590521498081990t^{7} + 56210703902675t^{6} + 4025955492130t^{5} + 192944303520t^{4} + 5186181089t^{3} + 89438568t^{2} + 1525136t + 1152 \right)D + 360 t ^{3} \left( 9318662109375000t^{17} + 18024436523437500t^{16} + 18039542578125000t^{15} + 11396701394531250t^{14} + 4956314203281250t^{13} + 2637079033062500t^{12} + 1822283716825000t^{11} + 997933947996875t^{10} + 377240853669375t^{9} + 97403477662225t^{8} + 17050568085885t^{7} + 2115751105603t^{6} + 192396296544t^{5} + 13071315465t^{4} + 551909436t^{3} + 13327812t^{2} + 272412t + 3888 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ4P4

|λ| = 10 λ ≈ 10; multiplicity 1; Jordan blocks 1 λ ≈ -2.5 - 7.694209i; multiplicity 1; Jordan blocks 1 λ ≈ -2.5 + 7.694209i; multiplicity 1; Jordan blocks 1 λ ≈ -2.5 - 1.816356i; multiplicity 1; Jordan blocks 1 λ ≈ -2.5 + 1.816356i; multiplicity 1; Jordan blocks 1 λ ≈ -1 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1 + 2i; multiplicity 2; Jordan blocks 1, 1 2 2
Sector dimension
9
Spectral radius
$10$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10$11
$-2.5 - 7.694209\mathrm{i}$11
$-2.5 + 7.694209\mathrm{i}$11
$-2.5 - 1.816356\mathrm{i}$11
$-2.5 + 1.816356\mathrm{i}$11
$-1 - 2\mathrm{i}$21, 1
$-1 + 2\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda - 10\right) \left(\lambda^{2} + 2 \lambda + 5\right)^{2} \left(\lambda^{4} + 10 \lambda^{3} + 100 \lambda^{2} + 375 \lambda + 625\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda - 10\right) \left(\lambda^{2} + 2 \lambda + 5\right) \left(\lambda^{4} + 10 \lambda^{3} + 100 \lambda^{2} + 375 \lambda + 625\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 15 & 15 & 15 & 40 & 40 & 30 \\ 1 & -1 & 0 & 0 & 0 & 4 & 21 & 18 & 12 \\ 2 & 3 & 0 & 0 & 0 & -2 & 12 & 6 & 24 \\ 0 & 2 & 1 & -1 & 0 & 0 & -2 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 & 4 & 0 & 9 \\ 0 & 0 & 2 & 3 & 4 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 2 & 0 & 3 & 2 & 4 & 0 \\ 0 & 0 & 0 & 1 & \frac{5}{2} & -1 & -\frac{3}{2} & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{10}{3} & \frac{10}{3} & 0\end{pmatrix}\notag\end{equation}$$