Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-297-65-a

·

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

  • $\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a quartic del Pezzo surface, a complete intersection of two quadrics.

  • $\pi_2\colon X\to\mathbb P^1$ is the fibration in the quadric threefolds of the pencil through the surface; over $5$ points the fibers are quadric cones of rank $4$.

Varieties in this family can be blown up to

Varieties in this family can be blown down to

Defining data
Ambient
ℙ4P4 × ℙ9P9
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_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11node 12: fundamental weight ω_12node 13: fundamental weight ω_13ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11ω12ω13
Bundle
E^(w1 + w5) + 2 copies of E^(w4 + w5)
Bundle
O_1(1) box O_2(1) + 2 copies of Q_1 box O_2(1)
ZL62 label
40900.w0Xw0Xw03
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;10) \\ \det\mathcal{E} &\cong \mathcal{O}(3;9) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 65
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 297
$\chi(X, \mathrm{T}_X)$ = -2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 297
$\mathrm{c}_1^2\mathrm{c}_2$ = 174
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 46
$\mathrm{c}_4$ = 13
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 7 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 + 2 0 x0 + 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 55 0 0 x1 - 25 80
$\mathrm{HH}^{ 4 }$ = 65 0 0 0 0 65
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 60t + 175t^{2} + 60t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(23)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(30)

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) \left( t + 1 \right) ^{2} \left( 8539t^{5} + 2914t^{4} + 214t^{3} - 2t^{2} - 2t - 1 \right) \left( 7175234115144t^{9} + 11445102302976t^{8} + 6843936616502t^{7} + 2366195539425t^{6} + 774779409132t^{5} + 126287905803t^{4} + 5897489778t^{3} + 456993387t^{2} + 1057860t + 66825 \right)D^{6} + 2 \left( t + 1 \right) \left( 643327903146753468t^{16} + 1130211739795805208t^{15} + 469215633442836395t^{14} - 278573821282783758t^{13} - 344925902204180157t^{12} - 181750190922378141t^{11} - 74401297705172997t^{10} - 19488372097369020t^{9} - 2779004224788267t^{8} - 245433875100738t^{7} - 22591356598677t^{6} - 3111804396192t^{5} - 373559394303t^{4} - 13897148301t^{3} - 910479069t^{2} - 1319490t - 66825 \right)D^{5} + \left( 10722131719112557800t^{17} + 28019113404256878840t^{16} + 26110037119919899518t^{15} + 8252175269354154087t^{14} - 2495531562048883219t^{13} - 3410568242732651106t^{12} - 1993492169174399190t^{11} - 806118438290910230t^{10} - 186021696748533775t^{9} - 22290111439459911t^{8} - 1579898771601454t^{7} - 70805738522037t^{6} + 5742493816575t^{5} + 1018424855544t^{4} + 26065864278t^{3} + 1705981716t^{2} + 2073195t + 66825 \right)D^{4} + 2 t \left( 22516476610136371380t^{16} + 56299335085042932060t^{15} + 52959195777356634745t^{14} + 21947359999844106901t^{13} + 2588236566493301205t^{12} - 1920415503437529183t^{11} - 1945122830465854341t^{10} - 915344846400130200t^{9} - 209339865712180914t^{8} - 23766877186156422t^{7} - 1894382883343323t^{6} - 118457815363635t^{5} - 2867688174777t^{4} - 25978963527t^{3} - 449881281t^{2} + 44550t - 12150 \right)D^{3} + 4 t ^{2} \left( 24875345588341134096t^{15} + 60049777123277757576t^{14} + 56324217669853802602t^{13} + 26155693449872875589t^{12} + 6925121622991385504t^{11} + 605058067946894649t^{10} - 951915503703874782t^{9} - 576285185096268312t^{8} - 131501786266019567t^{7} - 14073335709369744t^{6} - 1196257199068447t^{5} - 71641968376881t^{4} - 1508775693093t^{3} - 22861386525t^{2} - 453156201t + 23760 \right)D^{2} + 72 t ^{3} \left( 1501098440675758092t^{14} + 3528723078304937964t^{13} + 3283913541819892237t^{12} + 1608281425589574009t^{11} + 532780442728709717t^{10} + 114629888872505885t^{9} - 21736152506365873t^{8} - 21218064603119222t^{7} - 4868620946743554t^{6} - 500673077981442t^{5} - 45934534662525t^{4} - 2536268285907t^{3} - 44660211111t^{2} - 941715639t - 15757335 \right)D + 144 t ^{3} \left( 306346620546073080t^{14} + 707053343831060760t^{13} + 652787622036391898t^{12} + 328064686826956810t^{11} + 119716350853776245t^{10} + 31018595603850591t^{9} - 788253682354374t^{8} - 2889993880768248t^{7} - 669308615153289t^{6} - 67479666213042t^{5} - 6659177582925t^{4} - 335146749066t^{3} - 5008103640t^{2} - 136880685t - 1804275 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ4P4

|λ| = 8.181035 λ ≈ 8.181035; multiplicity 1; Jordan blocks 1 λ ≈ -0.7513808 - 7.191941i; multiplicity 1; Jordan blocks 1 λ ≈ -0.7513808 + 7.191941i; multiplicity 1; Jordan blocks 1 λ ≈ -4.339137 - 1.06458i; multiplicity 1; Jordan blocks 1 λ ≈ -4.339137 + 1.06458i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ 1; multiplicity 1; Jordan blocks 1 2
Sector dimension
8
Spectral radius
$8.18103506$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.181035$11
$-0.7513808 - 7.191941\mathrm{i}$11
$-0.7513808 + 7.191941\mathrm{i}$11
$-4.339137 - 1.06458\mathrm{i}$11
$-4.339137 + 1.06458\mathrm{i}$11
$-1$22
$1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda - 1\right) \left(\lambda^{5} + 2 \lambda^{4} + 2 \lambda^{3} - 214 \lambda^{2} - 2914 \lambda - 8539\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda - 1\right) \left(\lambda^{5} + 2 \lambda^{4} + 2 \lambda^{3} - 214 \lambda^{2} - 2914 \lambda - 8539\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 6 & 16 & 16 & 10 \\ 1 & -1 & 0 & 0 & 0 & 6 & 9 & 8 \\ 3 & 2 & 0 & 0 & 0 & 0 & 6 & 16 \\ 0 & 3 & 1 & -1 & 0 & 0 & 0 & 6 \\ 0 & 0 & 3 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$