1Segre cubic 4-foldShigeru MUKAI

6/17/22(F)@Roma Univ. Tre

Abstract: The cyclic triple covering of the projective 4-space with branch
the Segre cubic is characterized by 10 cusps as the Segre 3-fold is so by 10nodes. The Fano variety of lines is birationally equivalent to the Hilbertsquare of a K3 surface studied by Vinberg(1983). I will discuss thebinational automorphism group of this holomorphic symplectic 4-fold.

§1 Main motivation and side job§2 K3^[2] 4-folds§3 Vinberg's theorem and generalization to K3^[2] 4-folds§4 Review of Vinberg's proof§5 Proof of main theorem
References (8 items)§1 Main motivation and side job
Main motivation: generalize Aut(K3) to Bir-Aut(HK)(replacing nef cone with movable cone).
Side job: Kummer theory
l2Θl3(I start with) Kummer quartic surface, Φ: A/±1
P(1) A= J(C), C: curve of genus 2, Φ : embedding(16 6 - 16S ) configuration of nodes and tropes11(2) A = E 1× E 2,: Φ is of degree 2 onto a smooth quadric P × P2

τ = f4(x)g4 (y)branch locus

2Order 3 analogue(A, σ) : Abelian surface & symplectic automorphism of order 3
Quotient A/σ is a K3 surface with 9 cusps, 9A2
33(1) A=J(C), C: τ2= q(x ), σ: x → ωx, ω = 1(Barth-)Bertin-Vanhaecke sextic. A/σ 2[4P , Image = (2, 3) c.i.
(944- 9 ) configuration of 9 cusps and 9 conics
S(2) A = E × E, E with CM of order 3
A/σ → P P, τ 3 = (x2 - x)(y - y)IS-branch2

4[A/σ - P ] = (quadric cone)

I(3)c.i.

Mini history of Aut S and Picard lattice in U + (Leech)(1)
Kummer Kondo(1998), (A3+6A )ord 3 ver. Not yet, (A 5+ A I)
I(2)Keum-Kondo(2000), (2D )Vinberg(1983), E ⑥

§2. K3^[2] 4-folds
Simplest higher dimensional analogue of a K3 surface isthe Hilbert square of a K3 (and its deformations)LetS^[2] → S^(2) be the minimal resolution of symmetric product.
S^[2] = moduli of ideal sheaves of 2 points (allowing inf. nears)= MS(1, 0, -1)
Pic S^[2] = Pic S + Zδ, δ = (1, 0, 1)= (1, 0, -1) in Z+ Pic S +Z
Notation. α on S is identified with (0, α, 0) on S^[2].
Relation with cubic 4-folds and K3 sextic S = (2, 3), c.i. in P^4.
S : q(x, y, z, u, v) = d(x, y, z, u, v) = 0 in P^4
X = XS: q(x, y, z, u, v)w + d(x, y, z, u, v) = 0 in P^5This cubic 4-fold is singular at (100000).
Fact: Fano variety of lines F(XS) is birational with S^[2](1) S : Bertin-Vanhaecke ⇨ XS has A I + 9A2(2) S : Vinberg ⇨ XShas 10A 2 since the quadric hull of S is a cone
Observation: XS is a 4-dimensional analogue of Segre cubic 3-foldwhen S : Vinberg.
<1y5
Y:

22: 4yi
=

X :s=

0

(Ex)(2x?)

2=2?

<145

Segre cubic 3-fold Y is characterized by 10A I
4-fold X
10A 2XS is a triple cyclic covering of P^4 with branch Segre cubic.

3

4
Both have an action of symmetric group Jo of degree 6=C.E. (8, 3).22](SCI)
AntllArt(X, <MY)=

Normalizer of σ0.7 (123)(456)=: σ§3 Vinberg's theorem and generalization to K3^[2] 4-folds
Theorem (Vinberg)
Aut S = (Free product of 12 involutions) A Aut(S P^4)
CMain Theorem Bir-Aut(S^[2]) is semi-direct product of<84+120 involutions> by symmetric group En of degree 7.
Surprisingly,
56 -action on S extends to a birational En -action on S^[2].
Reason: a symmetry of degree 7 induced by Lagrangian fibration
S^[2] •••→ P^2(of 3A_6-type).
~Mordell-Weil group has a 7-torsion (birational) section.

52-1§4 Review of Vinberg's proof
TS =12Quick review on Aut S, Picard number 20,0 I
Pic S = U + 2 E_8 + A_2
U=, A_2 =10Orhtogonal group O (U+2E_8+A_2)= <24 (-2)-reflections, 12 (-6)-reflections> Aut(S < P^4)
24=6 + 9 •2-

36P^1's

9A_2's-~·~-----

Dual graph of 24 P^1's

6 circuits of length 18
S → P^1 elliptic fibration
with I_18 fiber, MW group Z + Z/6⇨ 2 anti-symplectic involutions(-1)-mult. &its composite with 2-torsion translation
Modulo Aut(SE P^4), these act as(-6)-reflection.

3-211-2

6§5 Proof of main theorem
Look at the action of Bir-Aut S^[2] on Pic S + Zδ, and on
the movable cone in it. The rest is basically the same as(Vinberg's) K3, but get complicated in two points:(1) the orthogonal group is no more more reflective(2) Divisibility should be taken into account. There are twotypes of (-2)-divisors:a) (-2) eﬀective divisor Im[E × S → S^[2]] for (-2)-curve
E on S ( divisibility 1)b) half δ of exceptional divisor class (divisibility 2)(1) is overcome by Conway-Borcherds domain CB in thepositive cone.
CB domain is surrounded by 309 walls:
309 = 35 + 70+ 84+ 120u↑↑(-2)-walls (-6)-walls (-42)-walls(divisibility 1 & 2,respectively)
↑ ↑↑↑
① ②③④

105 (-2)-reflections★ O (U+2E_8+A_2+A_1)= 84 (-6)-reflections
120 quasi-reflections
symmetry ofCB domain

57

7

Geometrization of ★
① Basic (-2)-divisors (of divsibility 1)# = 35 = 24 + 2 + 9

24 are "pull-back" of (-2)'s from S.
Extra 11(=2+9) are (1, f, 1) for elliptic pencil |f| on S of minimal
Coxter number (= 12). Since f is isotopic, this divisor has Beauvillesquare (-2). Geometrically, (1, f, 1) is the Zariski closure of locus of{a, b} with a = b and Φ(a) = Φ(b).
Two f's are of type 3E_6 and 9 of type D_7+A_11.

W3Es+ EnThe dual graph of these 35 (-2) divisors is the 4-valent odd graph
O_4.

8② (-2) divisors of divisibility 2 ⇄ 70 edges of O_4③ (-6)-walls ⇄ Induced automorphism from S④ (-42)-walls: Non-induced automorphism ⇄ (-1) multiplication of~

Lagrangian fibration of type 3A E(mod)Final answer: Aut S^[2] = <84+120 involutions>

Jy.

References(added in June 2023)§1 Main motivation and side job
Bertin, J. and Vanhaecke, P.: The even system and generalized
Kummer surfaces, Math. Proc. Camb. Phil. Soc., 116(1994),
131-142.
Barth, W.: K3 surfaces with nine cusps, Geom. Dedicata, 72(1998),
171-178.
Kondo, S.: The automorphism group of a generic Jacobian
Kummer surface, J. Alg. Geom. 7(1998), 589-609.
Keum, J-H. and Kondo, S.: The automorphism groups of Kummer
surfaces associated with the product of two elliptic curves, Trans.
Amer. Math. Soc., 353(2000), 1469-1487.
Vinberg, E.B.: The two most algebraic K3 surfaces, Math. Ann.
265(1983), 1-27.
§2 K3^[2] 4-folds
Dolgachev, I.: Corrado Segre and nodal cubic threefolds, in “Fromclassical to modern algebraic geometry; Corrado Segre’smastership and legacy”, Springer, 2016, pp. 429-450.
§3 Vinberg's theorem and generalization to K3^[2] 4-folds
Oguiso, K.: Picard number of the generic fiber of an abelian variety
fingered hyperkahler manifold, Math. Ann., 344(2009), 929-937.
§5 Proof of main theorem
Borcherds, R.: Automorphism groups of Lorentzian lattices,
J. Algebra, 111(1987), 133-153.

R