Fano 3-folds, Lagrangian fibration, and a supersingular OG10 withCo 2 configuration
11/9/22(W) Nagoya Univ.

ヒ

S.MuhaiSmsX--SEsSyiSRS@zS--P.X§1 Preliminary

K3 surface

min. resolution↑
1/2 of exceptional divisor
class= Hilbert scheme of n-pts|= moduli of ideal of colength n

Mg し v ) with v=(1, 0, 1-n)Isom. as latticeintersection form ←→ Beauville form q s.t. q^n ~ self-int #Lagrangian ﬁbration

ヨX

ピ"→gen. fiber= Lagrangian A.V.

>

'iflopl'

ヨDr、-_-Uで.

171か 。 B=た→T

B"(6)(p, q) → π(p)π(q)'lineindetermanacy

Dx.

(Bayer-Macri)

なるがD with q(D)=0<:

Basic Example S
刁 S={ fiber==h-S,た 桃川

JE で ( l )ヒ§2 Examples related with Fano 3-folds

S = S = (2, 2, 2) c が① Very general K3 of degree 8'Pie S=⊕区Zh(E) 88,818)

QがX

fbnctionover.net

2.

What is the Lagrangian fibration
朗of X=S for D = h - 2δ withq(D) = 0?

。

=Answer. や=> がJacobs昈
〈 Q Q Q >*on

is=,dfpadricsdefnir.gs:,,'=g.

In fact, Φ is O’Grady map. {p, q} E S → line pq 4 S g

いTUS g" いっ 、 isthemap

くヨ! V4{p, q}

1=QI がく が。

3-〈QI,2,*2

subpencil

fiber = Fano variety of lines in a fixed V4= Jacobian of curve of genus 2q.e.d.

② Very general K3 of degree 18, g= 10Q

両.

ERS

S ははく がWhat is the Lagrangian fibration of X=S for D = h - 3δwith q(D) = 0?
m彧Key variety:(of Borcea)=

GAは CB は) が=contact Fano manifold[Sはくが]=[]"。

Hi n Han H 3

Hint 1. ① and ② are similar.

Hint 2. S in ① is also a linear section:
8Sg

二[ v2 ( 昈 ) く が J 。 Hi 。 Han H 3
1111Sp (6)分D (s p (6 ))adj"

D ( S ピ)Moreover, a linear section of contact Fano 5-fold!
KEY: Rational homogeneous contact manifold

なX = G [ ] く なほ ).

simple Liealgebrav

h.w. vector

has a unique conic property: for every pair p, q EX in generalposition, unique conic C on X passing through p, q.

織っPf. Put p=[u], q=[v]. Then u and v generate a 3-dim’l Liesubalgebra a.
. C is the intersection with X and the 2-plane
㽗が "Answer.

・・-

SiC " wentand inFaw 3 Hd-The Laguna

inV18

f bin

きaUnitむ な 。 げ HI。

X がない か=.

.

seudlp.pt
Of 1か!

=B飴 こく Hi H,,the MrPencil=S8〈 Hi , 11 2 H , > dGoing S.

,fiber = Fano variety of conics in a fixed V= Int-Jac of V = Jacobian of curve of genus 2(Kuznetsov et al)

REMARK:

さ?

Conjecture. “contact Fano → homogeneous,”

which includes (Hartshorne’s conj.=) Mori’s theorem as specialcase.
Side Problem. Does a contact Fano manifold satisfy a uniqueconic property? (Here “conic” means a curve of degree 2 withrespect to the contact line bundle, whose (dim. - 1)/2-th power
is anti-canonical.)③ Very general K3 of degree 16, g= 9 (Omitted)eE§3 Leech-K3 analogue of del Pezzo surfaces
Similarity between I 1, 9 and II 1, 25EI」

Hyperbolic lattice, unimodular, odd and even
Both have beautiful fundamental domains & str. of orthog. grps

0 (Ia,,Z) ~{城Pby (い)りで W ( ED・詰っReflections

Symmetry of afund. domain

Coxeter group(Conway ‘80’s)

0 位は R),CAG (classical algebraic geometry)~|'by丿

Senated( と)refle.tns^

24 Cq
八八LeechlatticeO(Leech)I1, 9 contains Pic R d, the Picard lattice of a del Pezzo surface ofdegree d, as the orthogonal complement of the sum of d copiesof <-1>.
R d = Bl-up of the plane at 9-d points in general position

EThe nef cone is finite polyhedral, and the walls are defined bylines.

26Th "faeetsim.de!
d123

6→、でa.

EH.

!

一にc=sa風Pie R

順風cone○ mjn
Remark (1) -K is the Weyl vector in the sense that (-K. l)=1 for all
wall defining vectors l.
(2) The Weyl vector is isotropic for bothand-t.TT

d6 5 4 3 2
1 0# of lines 6 10 16 27 56 240 ∞root system ,d=1 graph of line configuration
vertexline
1I.
Fixed linel.

)

156

にて
71 26、Bertini involution of R

Intersection #(l56-101I3

Task: Geometrize.
in the framework

as possible as one cannegative definite
Leech lattice

ExtendedLeech latticeV

De ( 水 ) ↳ I

津区 ⊕。

Λ⊕ Rx,s)(r,()

where * is a K3 surface, K3-like object/
category, etc.
(x2)Λ - 2rs

quad. form

Leech analogy of degree 1 del Pezzo is the double Conway graphΓz{vertex (1, x, -1), both x and z-x have min. norm (# = 4600)suitable adjacency by intersection numberin the orthogonal complement of sum of two copies of <-2>, one is
generated by (1, 0, 1) and the other by (1, z, -1), for a fixed z of min.
norm.

11408

1408

1782
く〉

Bertini-like involution
x ←→ z-x1遝

8LThe double Conway graph has symmetry of the 2nd Conway
group Co2 The vertex stabilizer group is the unitary group
Sub-task: Find a K3-like object of Picard number 24 whichincarnates the double Conway graph.

§4. Relation between §2 and §3§3 is partly inspired by the unfinished/untreated case of §2,
namely the case of genus 8Case

①.

②.

③.

④

Sympl. Var. S^[2]S^[2] S^[3] OG10?
FanoQuartic dP g=10 g=9g=8」Partial answer: The Leech analogy of degree 1 del Pezzo surface
must be an OG10-like symplectic variety (with a Lagrangianfibration) related with theftp.
Fermat cubic 4-fold in characteristic 2.

Another supporting fact: the number 1782 in the graph is twice thenumber (=891) of 2-planes in the 4-fold.
I hope I will have another chance to discuss about this topic innear future.
Thank you for your attention, and the organizers, both Sho
Tanimoto and Shigeyuki Kondo, for the wonderful conference.

References(12/02/22)§1
Bayer-Macri, MMP for moduli of sheaves on K3s via wall-crossings:
nef and movable cones, Lagrangian fibrations, Invent. math., 2014.
§2
Kuznetsov-Prokhorov-Shramov, Hilbert schemes of lines and conics
and automorphism groups of Fano threefolds, Japanese J. Math., 2018.
Beauville, A., Holomorphic symplectic geometry: a problem list, in"Complex and diﬀerential geometry", pp. 49-63, Springer-Verlag, 2011.
(See Sect. 3 for the conjecture "contact+Fano implies homogeneous".)§3Conway, J.H., The automorphism group of the 26-dimensional even
unimodular Lorentzian lattice, J. Algebra, 1983. Chap. 27 of [SPLG].
Brouwer-Maldeghem, Strongly regular graphs, Camb. Univ. Press,2022. (Conway graphs are explained in Chap. 10.)§4
Laza-Sacca-Voisin, A hyper-Kahler compactification of the intermediate
Jacobian fibration associated with a cubic 4-fold, Acta Math., 2017.
Li-Pertushi-Zhao, Elliptic quintics on cubic fourfolds, O'Grady 10, andLagrangian fibration, Adv. in Math., 2022.
Edge, W.L., Permutation representations of a group of order
9196830720, J. London Math. Soc., 1970.
Dolgachev-Kondo, A supersingular K3 surface in characteristic 2 andthe Leech lattice, IMRN, 2003.

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