Family 2-157-38-a

← 2-160-39-f · 2-147-36-a →

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

$\pi_1\colon X\to Y$ is the blow-up of a smooth codimension-$2$ linear section $Y$ of $\mathrm{LG}(3,6)\subset\mathbb P^{13}$ (1-256-57-a) along a smooth sextic del Pezzo surface, a divisor of tridegree $(1,1,1)$ in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1\subset\mathrm{LG}(3,6)$.
$\pi_2\colon X\to\mathbb P^4$ maps birationally onto $\mathbb P^4$ (1-625-126-a) with center a sextic surface, the projection of an anticanonically embedded sextic del Pezzo surface, with three ordinary improper double points; the general fiber over the center is $\mathbb P^1$, and the fibers over the three double points are smooth quadric surfaces.

More details (pdf)

Varieties in this family can be blown down to

Fl(1, 3, 6) · 54.3210 (preferred model)

Defining data

Ambient: Fl(1, 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: $O (1, 0) \oplus U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (0, 1) \oplus Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes U_{3}^{\lor}$ O(1,0) + U_2^v ot U_3^v + O(0,1) + Q_1^v ot wedge^2 U_2^v ot U_3^v
ZL62 label: 54.3210

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,5) \\ \det\mathcal{E} &\cong \mathcal{O}(2,4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	38
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	157
$\chi(X, \mathrm{T}_X)$	=	-8

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	157
$\mathrm{c}_1^2\mathrm{c}_2$	=	130
$\mathrm{c}_2^2$	=	109
$\mathrm{c}_1\mathrm{c}_3$	=	44
$\mathrm{c}_4$	=	14

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		8		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 + 8	0	x₀ + 8	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 33	0	0	x₁ + 11	22
$\mathrm{HH}^{ 4 }$ = 38	0	0	0	0	38
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 33t + 89t^{2} + 33t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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( 2t + 1 \right) \left( 3t + 1 \right) \left( 17t^{3} - 33t^{2} - 12t + 1 \right) \left( 187t^{3} + 96t^{2} + 14t + 1 \right) \left( 408234988598376t^{13} + 1430840567126288t^{12} + 2118681193286164t^{11} + 1772707607441092t^{10} + 942246534530350t^{9} + 340503359203838t^{8} + 88614120819551t^{7} + 17700815745718t^{6} + 2857085639177t^{5} + 364325227547t^{4} + 31864078475t^{3} + 1390878862t^{2} + 4200403t + 117279 \right)D^{6} + \left( 163520157623033900304t^{22} + 654101625327813432984t^{21} + 800184220747758409056t^{20} - 445643845586724494852t^{19} - 2628777683238106756400t^{18} - 3968007280467325032178t^{17} - 3597222869227901390328t^{16} - 2271047729199194115791t^{15} - 1064135991012261933107t^{14} - 385010745087003358540t^{13} - 111043122622286468838t^{12} - 26264250345514994125t^{11} - 5198214349607615242t^{10} - 861894101959271179t^{9} - 116743975713034040t^{8} - 12334902702953247t^{7} - 969259918906363t^{6} - 56971042590202t^{5} - 2997395602304t^{4} - 163402211085t^{3} - 5587613492t^{2} - 10724745t - 234558 \right)D^{5} + \left( 1362667980191949169200t^{22} + 5548587293932665700680t^{21} + 8036897444057764742624t^{20} + 1941578337076307349412t^{19} - 9867349217998107361524t^{18} - 17114535002902600827266t^{17} - 15542153845545065961382t^{16} - 9509510548232506529299t^{15} - 4277722179169106807885t^{14} - 1488560474147622059914t^{13} - 416585763284675439487t^{12} - 96533583840266805048t^{11} - 18719982300222103935t^{10} - 2985343253659940820t^{9} - 373654656950318482t^{8} - 34070796579764678t^{7} - 2001871377994376t^{6} - 54675822200221t^{5} + 995177542612t^{4} + 139439021425t^{3} + 5553647356t^{2} + 16664369t + 117279 \right)D^{4} + t \left( 5723205516806186510640t^{21} + 23626610007980123348040t^{20} + 37530478849847289664800t^{19} + 22582632700489145636596t^{18} - 12641012814364971949088t^{17} - 36022613365373780513318t^{16} - 34797967467978664159788t^{15} - 21264349785691086364733t^{14} - 9399522453168760238201t^{13} - 3216529726890307629544t^{12} - 893326459990269339456t^{11} - 207293066330409836061t^{10} - 40249589638959270736t^{9} - 6349035085327871131t^{8} - 771054153852407200t^{7} - 66994115282145819t^{6} - 3755204030044063t^{5} - 115171974464764t^{4} - 1610228441180t^{3} - 28556691015t^{2} - 354630144t - 132219 \right)D^{3} + 2 t ^{2} \left( 6322779428090644145088t^{20} + 26374404622100277291264t^{19} + 44193912566583144423424t^{18} + 34692886817000910104432t^{17} + 4485548723767540114248t^{16} - 17927546269082562390354t^{15} - 20730677950680683803658t^{14} - 13137003440207647692699t^{13} - 5836105753885224888469t^{12} - 1998422571211191859873t^{11} - 557949517851631377615t^{10} - 130392644894615427766t^{9} - 25282184015135513705t^{8} - 3910585087696819911t^{7} - 455015436110012109t^{6} - 36883072067634790t^{5} - 1866574167486114t^{4} - 50121998292512t^{3} - 734781865832t^{2} - 17037684783t - 129335658 \right)D^{2} + 8 t ^{2} \left( 1716961655041855953192t^{20} + 7216267298657701313292t^{19} + 12481137555583888541808t^{18} + 11037549976375887937166t^{17} + 4153078426887586111592t^{16} - 1517823230915597409669t^{15} - 3011675993326249564542t^{14} - 2088541842593535100725t^{13} - 954780452659211055156t^{12} - 332224213900906423121t^{11} - 94218776322229206878t^{10} - 22296394805610739938t^{9} - 4326620599841058362t^{8} - 657923962814973830t^{7} - 73745389287461973t^{6} - 5627044417246065t^{5} - 260293263701520t^{4} - 6258301735578t^{3} - 101917284121t^{2} - 2314400572t - 12197016 \right)D + 24 t ^{2} \left( 233600225175762714720t^{20} + 986791830575922989520t^{19} + 1738701291613912148544t^{18} + 1632704654914871311736t^{17} + 795817600449728310608t^{16} + 50078259732965426346t^{15} - 214209058563479353926t^{14} - 176243030823487382388t^{13} - 84847858555479947268t^{12} - 30365551716282913803t^{11} - 8807625811323621309t^{10} - 2117182634588981304t^{9} - 412019161313448628t^{8} - 61831732855394866t^{7} - 6724250903677257t^{6} - 488325628939313t^{5} - 20982299722376t^{4} - 466719541161t^{3} - 8758716692t^{2} - 169247375t - 625488 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 6)

Sector dimension: 9
Spectral radius: $14.2344224$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$14.23442$	1	1
$-5.525473 - 5.734048\mathrm{i}$	1	1
$-5.525473 + 5.734048\mathrm{i}$	1	1
$-3$	1	1
$-2.949054$	1	1
$-2.680045$	1	1
$-2$	1	1
$-1$	1	1
$0.4456224$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{3} + 14 \lambda^{2} + 96 \lambda + 187\right) \left(\lambda^{3} - 12 \lambda^{2} - 33 \lambda + 17\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 54 & 27 & 36 & 32 & 104 & 70 \\ 1 & -1 & 4 & 0 & 0 & 0 & 6 & 24 & 16 \\ 1 & 1 & -1 & 16 & 10 & 8 & 12 & 39 & 16 \\ 0 & 1 & 1 & -2 & -1 & 2 & 0 & 0 & 6 \\ 0 & 1 & 0 & 4 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 2 & -2 & 4 & 16 & 6 \\ 0 & 0 & 0 & -3 & -5 & 4 & -5 & -6 & 8 \\ 0 & 0 & 0 & 4 & 3 & 1 & 2 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{13}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta