Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-70-21-c

·

Blow-up of the genus-eight Mukai fourfold along a K3 surface of degree $14$.

  • $\pi_1\colon X\to Y$ is the blow-up of the Mukai fourfold $Y$ of genus $8$, a codimension-four linear section of $\mathrm{Gr}(2,6)$ (1-224-51-a), along a K3 surface of genus $8$, a codimension-two linear section of $Y$.

  • $\pi_2\colon X\to\mathbb P^1$ is the flat pencil of hyperplane sections of $Y$ (1-224-51-a) through the K3 surface, with general fiber a prime Fano threefold of genus $8$ and degree $14$; for general $X$ exactly $48$ fibers have one ordinary double point.

This is a Fano fourfold of K3 type, appearing as case K3-25 in Bernardara–Fatighenti–Manivel–Tanturri.

Varieties in this family can be blown down to

Defining data
Ambient
ℙ1P1 × Gr(2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
4 copies of E^(w3) + E^(w1 + w3)
Bundle
4 copies of O_2(1) + O_1(1) box O_2(1)
ZL62 label
151.2222h
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 70
$\chi(X, \mathrm{T}_X)$ = -27

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 70
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 138
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 36
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 28 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 + 27 0 x0 + 27 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 9 0 0 x1 - x2 - 12 0 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 36t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

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( 4t + 1 \right) ^{2} \left( 5t + 1 \right) ^{3} \left( 29t^{2} + 10t + 1 \right) \left( 421t^{2} - 46t + 1 \right) \left( 45785924016t^{9} + 37539560820t^{8} + 9519019924t^{7} - 272781172t^{6} - 565824776t^{5} - 102420123t^{4} - 6243916t^{3} + 23098t^{2} + 6944t - 15 \right)D^{6} + \left( 4t + 1 \right) \left( 5t + 1 \right) ^{2} \left( 234780145450764480t^{15} + 312394367947151232t^{14} + 164007555860750252t^{13} + 36893612597312232t^{12} - 813688412965088t^{11} - 2518268796909960t^{10} - 628649808638781t^{9} - 71554231924962t^{8} - 3051117496536t^{7} + 106654474080t^{6} + 12169968426t^{5} + 235848476t^{4} + 12815808t^{3} + 22888t^{2} - 20577t + 30 \right)D^{5} + \left( 5t + 1 \right) \left( 39130024241794080000t^{17} + 64199966896708660800t^{16} + 44018728226808762672t^{15} + 15266404205501382372t^{14} + 2023481491829547356t^{13} - 466613123884830996t^{12} - 274028548104580680t^{11} - 59971972699862249t^{10} - 7310434748687914t^{9} - 482845979664357t^{8} - 10802285764906t^{7} + 501712964204t^{6} + 19597892066t^{5} - 1182334376t^{4} - 68904654t^{3} - 175735t^{2} + 12412t - 15 \right)D^{4} + t \left( 821730509077675680000t^{17} + 1422219918592122984000t^{16} + 1043080367545063027600t^{15} + 398172045626474375156t^{14} + 66203432725884638608t^{13} - 8685489857006922964t^{12} - 7547478297924006548t^{11} - 1980628126654392879t^{10} - 297245979664217164t^{9} - 26943490426088123t^{8} - 1323016210742952t^{7} - 15589999212122t^{6} + 1536202075928t^{5} + 59284926750t^{4} + 140095500t^{3} - 10094475t^{2} + 347204t + 81 \right)D^{3} + 4 t ^{2} \left( 453908281204811328000t^{16} + 747433921389846052800t^{15} + 515697460238874353360t^{14} + 179553176453220645058t^{13} + 22621579406721445716t^{12} - 6882349270393706913t^{11} - 3934205617662355192t^{10} - 915519855516732500t^{9} - 125037792418147145t^{8} - 10248247090321493t^{7} - 435646179127725t^{6} - 1858096930866t^{5} + 618759401974t^{4} + 20797467548t^{3} + 140505615t^{2} + 487198t + 50565 \right)D^{2} + 4 t ^{2} \left( 493038305446605408000t^{16} + 781482131112085358400t^{15} + 513841532378337320400t^{14} + 165518777535545910312t^{13} + 15067776062531749320t^{12} - 8633075128495127070t^{11} - 4016114035020309588t^{10} - 857219020429818115t^{9} - 108484658931637340t^{8} - 8133949477065661t^{7} - 297938584289052t^{6} + 1337996257242t^{5} + 520059804048t^{4} + 15149604193t^{3} + 108026276t^{2} + 1329835t + 15600 \right)D + 96 t ^{2} \left( 8385005194670160000t^{16} + 12941387943873228000t^{15} + 8219494533530363640t^{14} + 2493940805045362344t^{13} + 157772089074397324t^{12} - 156866959785246556t^{11} - 64930832942751428t^{10} - 13048246294477705t^{9} - 1558320621881696t^{8} - 108597077906567t^{7} - 3450378193412t^{6} + 47639529630t^{5} + 7254877636t^{4} + 190087501t^{3} + 1384204t^{2} + 21385t + 100 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 6)

|λ| = 33.3923 λ ≈ 33.3923; multiplicity 1; Jordan blocks 1 λ ≈ 12.6077; multiplicity 1; Jordan blocks 1 λ ≈ -5 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -5 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 3; Jordan blocks 1, 2 λ ≈ -4; multiplicity 2; Jordan blocks 1, 1 3 2
Sector dimension
9
Spectral radius
$33.3923048$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$33.3923$11
$12.6077$11
$-5 - 2\mathrm{i}$11
$-5 + 2\mathrm{i}$11
$-5$31, 2
$-4$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{3} \left(\lambda^{2} + 10 \lambda + 29\right) \left(\lambda^{2} - 46 \lambda + 421\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right)^{2} \left(\lambda^{2} + 10 \lambda + 29\right) \left(\lambda^{2} - 46 \lambda + 421\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 72 & 192 & 426 & 774 & 1360 & 3960 & 11400 \\ 1 & -1 & 4 & 64 & 42 & 78 & 360 & 600 & 2600 \\ 1 & 1 & 5 & 18 & 56 & 102 & 150 & 510 & 1360 \\ 0 & 1 & 1 & 4 & 3 & 6 & 50 & 70 & 360 \\ 0 & 0 & 1 & 1 & -4 & 0 & 0 & 10 & 30 \\ 0 & 0 & 1 & 1 & 5 & 5 & 10 & 40 & 90 \\ 0 & 0 & 0 & \frac{14}{5} & 1 & \frac{9}{5} & 4 & 9 & 64 \\ 0 & 0 & 0 & 0 & 1 & \frac{9}{5} & 1 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$