Family 2-209-48-a

← 2-211-49-a · 2-208-48-a →

Blow-up of a codimension-two linear section of $\operatorname{LG}(3,6)$ along a quadric surface.

$\pi_1\colon X\to D$ is the blow-up of a Mukai fourfold of genus $9$ (index $2$, degree $16$), a general codimension-two linear section of $\mathrm{LG}(3,6)\subset\mathbb P^{13}$ (1-256-57-a), along a quadric surface $\mathbb P^1\times\mathbb P^1$.
$\pi_2\colon X\to W$ is the blow-up of the del Pezzo fourfold of degree $5$, a linear section of $\mathrm{Gr}(2,5)$ (1-405-85-a), along an anticanonically embedded sextic del Pezzo surface of class $2\sigma_2$.

More details (pdf)

Varieties in this family can be blown down to

Fl(2, 3, 6) · 55.3210 (preferred model)

Defining data

Ambient: Fl(2, 3, 6)
Bundle: $E^{ω_{1}} \oplus E^{ω_{2}} \oplus E^{ω_{3}} \oplus E^{ω_{4}}$ E^(w1) + E^(w2) + E^(w3) + E^(w4)
Bundle: $U_{3}^{\lor} \oplus O (1, 0) \oplus O (0, 1) \oplus Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor}$ U_3^v + O(1,0) + O(0,1) + Q_1^v ot U_2^v ot wedge^2 U_3^v
ZL62 label: 55.3210

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	48
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	209
$\chi(X, \mathrm{T}_X)$	=	-4

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	209
$\mathrm{c}_1^2\mathrm{c}_2$	=	146
$\mathrm{c}_2^2$	=	103
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	12

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		6		0		0
	0		0		0		0
		0		2		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ + 4	0	x₀ + 4	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 41	0	0	x₁ - 1	42
$\mathrm{HH}^{ 4 }$ = 48	0	0	0	0	48
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$

$\displaystyle\frac{ 1 + 43t + 121t^{2} + 43t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching 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( t - 1 \right) \left( t + 1 \right) \left( 2t + 1 \right) \left( 1607t^{6} - 1548t^{5} - 194t^{4} + 731t^{3} + 111t^{2} - 4t - 1 \right) \left( 6049324970824t^{13} + 10929334018064t^{12} + 11989998855204t^{11} + 9208456972396t^{10} + 2593982497034t^{9} - 1060570502598t^{8} - 1043288512785t^{7} - 180050913726t^{6} + 51075738249t^{5} + 29989081583t^{4} - 2101478729t^{3} + 95931126t^{2} + 3911003t + 17347 \right)D^{6} + \left( 408293139580795056t^{22} + 619664786762801400t^{21} + 188394378225571960t^{20} - 266808602791196996t^{19} - 603324528484202820t^{18} - 382491299144402694t^{17} + 256169407348833138t^{16} + 407099354908832893t^{15} + 114798532724237138t^{14} - 63871229818751230t^{13} - 40035282597207387t^{12} + 3602091494400935t^{11} + 7891393221924896t^{10} + 1147123475670125t^{9} - 638658859896690t^{8} - 132569350804239t^{7} + 7242557873636t^{6} - 856051443704t^{5} - 241271762003t^{4} + 8307314355t^{3} - 371083014t^{2} - 11524845t - 34694 \right)D^{5} + \left( 3402442829839958800t^{22} + 5638250950091914600t^{21} + 3839020469231443272t^{20} + 373916418202366196t^{19} - 4237266102451634288t^{18} - 4360141454668881434t^{17} - 154611450762533816t^{16} + 1899032671334286351t^{15} + 911054893053530826t^{14} - 116101501647297222t^{13} - 190041291568109210t^{12} - 27140972894735173t^{11} + 19225906996570481t^{10} + 2180459110700140t^{9} - 2350831750224964t^{8} - 223729226492200t^{7} + 28378953406629t^{6} - 2071428791199t^{5} + 127942063607t^{4} - 15365625386t^{3} + 408839572t^{2} + 9876671t + 17347 \right)D^{4} + t \left( 14290259885327826960t^{21} + 25246100489712321000t^{20} + 23032389136446921320t^{19} + 10704540857290323124t^{18} - 12032976675200843580t^{17} - 18108012087215949730t^{16} - 4975694325446635710t^{15} + 4196067661026620043t^{14} + 2868211407930714802t^{13} - 57064329952921846t^{12} - 562201893925145223t^{11} - 151852199401839703t^{10} + 33781482819475394t^{9} + 3537223420801641t^{8} - 4088683581192326t^{7} + 15736394608725t^{6} + 34493990589066t^{5} - 6399664426042t^{4} - 306216944453t^{3} - 6693350613t^{2} - 38231146t - 7719 \right)D^{3} + 2 t ^{2} \left( 15787334730457408832t^{20} + 29214105082112729120t^{19} + 30925396381716754880t^{18} + 19626994148342696624t^{17} - 7342731538617458928t^{16} - 18160942470192316022t^{15} - 7344090432436379864t^{14} + 2559473423496291463t^{13} + 2558744195343968590t^{12} + 198189565417287131t^{11} - 441450741857996999t^{10} - 142082032416644544t^{9} + 26387970583811689t^{8} + 2543181932820177t^{7} - 1973088649455979t^{6} + 169068204297944t^{5} - 1092326941477t^{4} - 3729581801548t^{3} - 146255927860t^{2} - 2649537753t - 15831332 \right)D^{2} + 8 t ^{2} \left( 4287077965598348088t^{20} + 8196450913691193900t^{19} + 9438817884229254780t^{18} + 6869310461500951190t^{17} - 738859412747851050t^{16} - 4430419993515105843t^{15} - 2178563917404805971t^{14} + 411655426034344284t^{13} + 609180584122632819t^{12} + 100138480491750880t^{11} - 88280832142481268t^{10} - 30618512070521394t^{9} + 6313494522831934t^{8} + 516266234903856t^{7} - 248738268495981t^{6} + 41533371308505t^{5} - 3213931032219t^{4} - 532675441893t^{3} - 20442319036t^{2} - 295078221t - 1665312 \right)D + 8 t ^{2} \left( 1749827741060550240t^{20} + 3418099004629350000t^{19} + 4131300662684199216t^{18} + 3224022055056695256t^{17} + 49383353385114416t^{16} - 1656840760214673382t^{15} - 916317658138539612t^{14} + 108353978889012744t^{13} + 230808273264378078t^{12} + 52953062239163847t^{11} - 28191150306523094t^{10} - 10147342232251136t^{9} + 2417092834373877t^{8} + 163520334507634t^{7} - 50195444221922t^{6} + 13118569687011t^{5} - 1490260367583t^{4} - 123133801779t^{3} - 5014480224t^{2} - 58781347t - 277552 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 6)

Sector dimension: 9
Spectral radius: $11.3096405$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$11.30964$	1	1
$-7.591141 - 2.262038\mathrm{i}$	1	1
$-7.591141 + 2.262038\mathrm{i}$	1	1
$-2$	1	1
$-1.981098$	1	1
$0.9268699 - 0.5329696\mathrm{i}$	1	1
$0.9268699 + 0.5329696\mathrm{i}$	1	1
$-1$	1	1
$1$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda - 1\right) \left(\lambda^{6} + 4 \lambda^{5} - 111 \lambda^{4} - 731 \lambda^{3} + 194 \lambda^{2} + 1548 \lambda - 1607\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 3 & 12 & 6 & 8 & 16 & 20 \\ 1 & -1 & 2 & 4 & 0 & 4 & 6 & 6 & 0 \\ 1 & 1 & -1 & 2 & 8 & 4 & 0 & 0 & 8 \\ 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 3 & 1 & -1 & 1 & 4 & 4 & 0 \\ 0 & 2 & 1 & -1 & 1 & -1 & 0 & 4 & 6 \\ 0 & 0 & 0 & 1 & 7 & 2 & -1 & 2 & 8 \\ 0 & 0 & 0 & 2 & \frac{1}{2} & \frac{5}{2} & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta