Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-513-105-b

·

The projective bundle $\mathbb P_{\mathbb P^2}(\mathcal O\oplus\mathcal O(-1)^{\oplus2})$.

  • $\pi_1\colon X\to\mathrm{Gr}(2,5)$ maps birationally onto a Schubert variety of degree $3$, the cone over $\mathbb P^2\times\mathbb P^1$; it contracts a plane with normal bundle $\mathcal O(-1)^{\oplus 2}$ to the vertex.

  • $\pi_2\colon X\to\mathbb P^2$ is the $\mathbb P^2$-bundle $\mathbb P_{\mathbb P^2}(\mathcal O\oplus\mathcal O(-1)^{\oplus 2})$.

Varieties in this family can be blown up to

Defining data
Ambient
ℙ2P2 × Gr(2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w2 + w3)
Bundle
Q_1 box U_2^v
ZL62 label
2041.l
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 105
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 513
$\chi(X, \mathrm{T}_X)$ = 18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 513
$\mathrm{c}_1^2\mathrm{c}_2$ = 222
$\mathrm{c}_2^2$ = 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
$\mathrm{HH}^{ 1 }$ = 18 0 18
$\mathrm{HH}^{ 2 }$ = 93 0 0 93
$\mathrm{HH}^{ 3 }$ = 172 0 0 0 172
$\mathrm{HH}^{ 4 }$ = 105 0 0 0 0 105
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 100t + 311t^{2} + 100t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(6)
Coates et al. (2020) Obro(4,47)
Coates–Kasprzyk–Prince (2015) CKP(10)

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( 85219t^{9} - 110592t^{8} + 46656t^{7} - 2187t^{6} - 512t^{5} + 432t^{4} + 81t^{3} + t - 1 \right) \left( 27076237983744t^{11} - 98209019068416t^{10} + 123897875571072t^{9} - 126142773329920t^{8} + 77025157662720t^{7} - 9592847330112t^{6} - 7297505596647t^{5} + 3016526510592t^{4} - 291067700892t^{3} + 14813991126t^{2} - 84626208t + 4487724 \right)D^{6} + \left( 48455608419470278656t^{20} - 235029097000479227904t^{19} + 454767289875158251392t^{18} - 583015928761041027072t^{17} + 531895009850980546560t^{16} - 296972797104109400832t^{15} + 70416368977020810633t^{14} + 16206992550765819904t^{13} - 15484321657328226756t^{12} + 4113670826022900255t^{11} - 413571117660724512t^{10} - 13243331350528060t^{9} + 7848403472770359t^{8} + 652872073086816t^{7} - 112873286285145t^{6} - 70982963900379t^{5} + 24980696488692t^{4} - 2064193843344t^{3} + 88968572964t^{2} - 423131040t + 17950896 \right)D^{5} + \left( 403796736828918988800t^{20} - 1934519896014650081280t^{19} + 3640040076945259170432t^{18} - 4673773247011573989376t^{17} + 4251995278686121893888t^{16} - 2249796400820473981248t^{15} + 432200007401609193183t^{14} + 179629177000385941504t^{13} - 130502978465971494900t^{12} + 32096988735333914163t^{11} - 3364082844672780576t^{10} + 73481461399828160t^{9} + 41761943846293288t^{8} - 6462623617875360t^{7} + 343682176275549t^{6} + 155995611663978t^{5} - 46252871783292t^{4} + 3287247415734t^{3} - 132263456742t^{2} + 316083708t - 13463172 \right)D^{4} + t \left( 1695946294681459752960t^{19} - 8045599052596696842240t^{18} + 14779525776633557850240t^{17} - 19034480681951807668224t^{16} + 17211431434701721257984t^{15} - 8563013339300029687680t^{14} + 1202709556529143038063t^{13} + 940626582730467878912t^{12} - 562929488209812212268t^{11} + 130071382126251843213t^{10} - 13655372573472923232t^{9} + 656409741396474996t^{8} + 46798598906387541t^{7} - 9825049607375904t^{6} + 615781128966993t^{5} + 14522842079946t^{4} + 156053096316t^{3} + 17172818556t^{2} + 512352t - 17496 \right)D^{3} + t ^{3} \left( 3747233717772368216064t^{17} - 17642745043423102107648t^{16} + 31777669814737133327616t^{15} - 41076954199828030160896t^{14} + 36868167011819052902400t^{13} - 17225547557751857491200t^{12} + 1492459923863548243494t^{11} + 2440715522043299587584t^{10} - 1276359147348050709384t^{9} + 280659981338658611490t^{8} - 29178375837901370496t^{7} + 1646515294041962592t^{6} + 2939516999058823t^{5} - 9195948030418944t^{4} + 1046417953766706t^{3} + 1853530138692t^{2} + 443860205544t + 13813214472 \right)D^{2} + 18 t ^{3} \left( 226126172624194633728t^{17} - 1058713924824664768512t^{16} + 1878231331592077902336t^{15} - 2436414998113829126144t^{14} + 2171383655745642639360t^{13} - 958248157212528009984t^{12} + 34637869219694373468t^{11} + 164716058302059400704t^{10} - 78562273014742310736t^{9} + 16626828042349466700t^{8} - 1712581386033172608t^{7} + 100645437145131000t^{6} - 2229076952165675t^{5} - 75083457720192t^{4} + 39609144544104t^{3} - 166295877120t^{2} + 24603155136t + 242337096 \right)D + 288 t ^{4} \left( 5768524811841699840t^{16} - 26905729747556597760t^{15} + 47229389894546009856t^{14} - 61434534719932235776t^{13} + 54449909963877200640t^{12} - 22978051550800933248t^{11} - 111989771342504910t^{10} + 4516717569598225152t^{9} - 2030933342453524038t^{8} + 418363311613268508t^{7} - 42790426755925608t^{6} + 2523952355123772t^{5} - 78286378623039t^{4} + 4930869275568t^{3} + 581420485998t^{2} - 1989753984t + 542515968 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 5.745649 λ ≈ 5.745649; multiplicity 1; Jordan blocks 1 λ ≈ -1.153163 - 5.35408i; multiplicity 1; Jordan blocks 1 λ ≈ -1.153163 + 5.35408i; multiplicity 1; Jordan blocks 1 λ ≈ -4.287135 - 1.86675i; multiplicity 1; Jordan blocks 1 λ ≈ -4.287135 + 1.86675i; multiplicity 1; Jordan blocks 1 λ ≈ 1.780116 - 2.833424i; multiplicity 1; Jordan blocks 1 λ ≈ 1.780116 + 2.833424i; multiplicity 1; Jordan blocks 1 λ ≈ 1.287357 - 0.6020321i; multiplicity 1; Jordan blocks 1 λ ≈ 1.287357 + 0.6020321i; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$5.74564939$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$5.745649$11
$-1.153163 - 5.35408\mathrm{i}$11
$-1.153163 + 5.35408\mathrm{i}$11
$-4.287135 - 1.86675\mathrm{i}$11
$-4.287135 + 1.86675\mathrm{i}$11
$1.780116 - 2.833424\mathrm{i}$11
$1.780116 + 2.833424\mathrm{i}$11
$1.287357 - 0.6020321\mathrm{i}$11
$1.287357 + 0.6020321\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\lambda^{9} - \lambda^{8} - 81 \lambda^{6} - 432 \lambda^{5} + 512 \lambda^{4} + 2187 \lambda^{3} - 46656 \lambda^{2} + 110592 \lambda - 85219\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{9} - \lambda^{8} - 81 \lambda^{6} - 432 \lambda^{5} + 512 \lambda^{4} + 2187 \lambda^{3} - 46656 \lambda^{2} + 110592 \lambda - 85219\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 3 & 0 & 4 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & -4 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 \\ 0 & 3 & 4 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 3 & 0\end{pmatrix}\notag\end{equation}$$