Moduli of supersingular K3 surfaces in characteristic 2 Ichiro - PDF document

Moduli of supersingular K3 surfaces in characteristic 2 Ichiro Shimada (Hokkaido University, Sapporo, JAPAN) § 1. Construction of the moduli space § 2. Stratification by codes § 3. Geometry of splitting curves and codes § 4. The case of Artin invariant 2 § 5. Cremona transformations We work over an algebraically closed field k of charac- teristic 2. 1

§ 1. Construction of the Moduli Space Let X be a supersingular K 3 surface. Let L be a line bundle on X with L 2 = 2. We say that L is a polarization of type ( ♯ ) if the following conditions are satisfied: • the complete linear system |L| has no fixed compo- nents, and • the set of curves contracted by the morphism Φ |L| : X → P 2 defined by |L| consists of 21 disjoint ( − 2)-curves. If ( X, L ) is a polarized supersingular K 3 surface of type ( ♯ ), then Φ |L| : X → P 2 is purely inseparable. Every supersingular K 3 surface has a polarization of type ( ♯ ). We will construct the moduli space M of polarized su- persingular K 3 surfaces of type ( ♯ ). 2

Let G = G ( X 0 , X 1 , X 2 ) be a non-zero homogeneous polynomial of degree 6. We can define Γ( P 2 , Ω 1 ∈ dG P 2 (6)) , because we are in characteristic 2 and we have O P 2 (6) ∼ = O P 2 (3) ⊗ 2 . We put { ∂G } = ∂G = ∂G P 2 . Z ( dG ) := { dG = 0 } = ⊂ = 0 ∂X 0 ∂X 1 ∂X 2 If dim Z ( dG ) = 0, then length O Z ( dG ) = c 2 (Ω 1 P 2 (6)) = 21 . We put U := { G | Z ( dG ) is reduced of dimension 0 } ⊂ H 0 ( P 2 , O P 2 (6)) . For G ∈ U , we put Y G := { W 2 = G ( X 0 , X 1 , X 2 ) } π G P 2 , − → and let ρ G : X G → Y G be the minimal resolution of Y G . We have Sing( Y G ) = π − 1 G ( Z ( dG )) = { 21 ordinary nodes } . 3

We then put L G := ( π G ◦ ρ G ) ∗ O P 2 (1) . ( X, L ) is a polarized supersingular K 3 surface of type ( ♯ ) ⇕ there exists G ∈ U such that ( X, L ) ∼ = ( X G , L G ) We put V := H 0 ( P 2 , O P 2 (3)) . Because we have d ( G + H 2 ) = dG for H ∈ V , the additive group V acts on the space U by G + H 2 ∈ U . ( G, H ) ∈ U × V �→ Let G and G ′ be homogeneous polynomials in U . Then the following conditions are equivalent: (i) Y G and Y G ′ are isomorphic over P 2 , (ii) Z ( dG ) = Z ( dG ′ ), and (iii) there exist c ∈ k × and H ∈ V such that G ′ = c G + H 2 . 4

Therefore the moduli space M of polarized supersingu- lar K 3 surfaces of type ( ♯ ) is constructed by M = PGL (3 , k ) \ P ∗ ( U / V ) . We put P := { P 1 , . . . , P 21 } , on which the full symmetric group S 21 acts from left. We denote by G the space of all injective maps → P 2 γ : P ֒ such that there exists G ∈ U satisfying γ ( P ) = Z ( dG ). Then we can construct M by M = PGL (3 , k ) \G /S 21 . Example by Dolgachev-Kondo: G DK := X 0 X 1 X 2 ( X 3 0 + X 3 1 + X 3 2 ) , Z ( dG DK ) = P 2 ( F 4 ) . The Artin invariant of the supersingular K 3 surface X G DK is 1. [ G DK ] ∈ M : the Dolgachev-Kondo point. 5

§ 2. Stratification by Isomorphism Classes of Codes Let G be a polynomial in U . NS ( X G ) : the N´ eron-Severi lattice of X G , disc NS ( X G ) = − 2 2 σ ( X G ) , ( σ ( X G ) is the Artin invariant of X G ). → P 2 be an injective map such that Let γ : P ֒ γ ( P ) = Z ( dG ) = π G (Sing Y G ) , that is, γ is a numbering of the singular points of Y G . E i ⊂ X G : the ( − 2)-curve that is contracted to γ ( P i ). Then NS ( X G ) contains a sublattice   − 2     − 2     S 0 = 〈 [ E 1 ] , . . . , [ E 21 ] , [ L G ] 〉 = .       − 2   2 S ∨ = Hom( S 0 , Z ) = 〈 [ E 1 ] / 2 , . . . , [ E 21 ] / 2 , [ L G ] / 2 〉 0 ⊃ NS ( X G ) . 6

We put � ∨ /S 0 = F ⊕ 21 C G := NS ( X G ) /S 0 ⊂ ⊕ F 2 , S 0 2 ∼ C G := pr( � F ⊕ 21 = 2 P (the power set of P ) . C G ) ⊂ 2 ∼ = 2 P is given by Here the identification F ⊕ 21 2 v �→ { P i ∈ P | the i -th coordinate of v is 1 } . We have dim � C G = dim C G = 11 − σ ( X G ) . We say that a reduced irreducible curve C ⊂ P 2 splits in X G if the proper transform of C in X G is non-reduced, that is, of the form 2 F C , where F C ⊂ X G is a reduced curve in X G . We say that a reduced curve C ⊂ P 2 splits in X G if every irreducible component of C splits in X G . 7

C ⊂ P 2 : a curve of degree d splitting in X G , m i ( C ) : the multiplicity of C at γ ( P i ) ∈ Z ( dG ). 21 ∑ 1 [ F C ] = 2( d · [ L G ] − m i ( C )[ E i ]) ∈ NS ( X G ) , i =1 � ∈ C G = NS ( X G ) /S 0 , w ( C ) := [ F C ] mod S 0 ˜ w ( C ) := pr( ˜ w ( C )) = { P i ∈ P | m i ( C ) is odd } ∈ C G . A general member Q of the linear system 〈 ∂G 〉 , ∂G , ∂G |I Z ( dG ) (5) | = ∂X 0 ∂X 1 ∂X 2 splits in X G . In particular, w ( Q ) = P = (1 , 1 , . . . , 1) ∈ C G . What kind of codes can appear as C G for some G ∈ U ? 8

NS ( X G ) has the following properties; • type II (that is, v 2 ∈ Z for any v ∈ NS ( X G ) ∨ ), • there are no u ∈ NS ( X G ) such that u · [ L G ] = 1 and u 2 = 0 (that is, |L G | is fixed component free), and • if u ∈ NS ( X G ) satisfies u · [ L G ] = 0 and u 2 = − 2, then u = [ E i ] or − [ E i ] for some i (that is, Sing Y G consists of 21 ordinary nodes). C G has the following properties; • P = (1 , 1 , . . . , 1) ∈ C G , and • | w | ∈ { 0 , 5 , 8 , 9 , 12 , 13 , 16 , 21 } for any w ∈ C G . The isomorphism classes [ C ] of codes C ⊂ F ⊕ 21 = 2 P 2 satisfying these conditions are classified: σ = 11 − dim C , r ( σ ) = the number of the isomorphism classes. σ 1 2 3 4 5 6 7 8 9 10 total 193 . r ( σ ) 1 3 13 41 58 43 21 8 3 1 9

the isomorphism class of ( X G , L G ) ∈ M [ C ] ⇐ ⇒ C G ∈ [ C ] ⊔ M = PGL (3 , k ) \ P ∗ ( U / V ) = M [ C ] . the isom. classes Each M [ C ] is non-empty. dim M [ C ] = σ − 1 = 10 − dim C . Case of σ = 1. There exists only one isomorphism class [ C DK ] with di- mension 10. P ∼ = P 2 ( F 4 ) , 2 P . C DK := 〈 L ( F 4 ) | L : F 4 -rational lines 〉 ⊂ The weight enumerator of C DK is 1 + 21 z 5 + 210 z 8 + 280 z 9 + 280 z 12 + 210 z 13 + 21 z 16 + z 21 . The 0-dimensional stratum M DK consists of a single point [( X DK , L DK )], where X DK is the resolution of W 2 = X 0 X 1 X 2 ( X 3 0 + X 3 1 + X 3 2 ) . 10

§ 3. Geometry of Splitting Curves and Codes G ∈ U . We fix a bijection ∼ γ : P → Z ( dG ) = π G (Sing Y G ) . Let L ⊂ P 2 be a line. L splits in ( X G , L G ), ⇐ ⇒ | L ∩ Z ( dG ) | ≥ 3, ⇐ ⇒ | L ∩ Z ( dG ) | = 5. Let Q ⊂ P 2 be a non-singular conic curve. Q splits in ( X G , L G ), ⇐ ⇒ | Q ∩ Z ( dG ) | ≥ 6, and ⇐ ⇒ | Q ∩ Z ( dG ) | = 8. The word w ( L ) = γ − 1 ( L ∩ Z ( dG )) of a splitting line L is of weight 5. The word w ( Q ) = γ − 1 ( Q ∩ Z ( dG )) of a splitting non- singular conic curve Q is of weight 8. 11

A pencil E of cubic curves in P 2 is called a regular pencil if the following hold: • the base locus Bs( E ) consists of distinct 9 points, and • every singular member has only one ordinary node. We say that a regular pencil E splits in ( X G , L G ) if every member of E splits in ( X G , L G ). Let E be a regular pencil of cubic curves spanned by E 0 and E ∞ . Let H 0 = 0 and H ∞ = 0 be the defining equations of E 0 and E ∞ , respectively. Then E splits in ( X G , L G ) if and only if Z ( dG ) = Z ( d ( H 0 H ∞ )) , or equivalently Y G and Y H 0 H ∞ are isomorphisc over P 2 , or equivalently ∃ c ∈ k × , H 0 H ∞ = cG + H 2 . ∃ H ∈ V , If E splits in ( X G , L G ), then Bs( E ) is contained in Z ( dG ), and w ( E t ) = γ − 1 (Bs( E )) holds for every member E t of E . In particular, the word w ( E t ) is of weight 9. 12

Let A be a word of C G . (i) We say that A is a linear word if | A | = 5. (ii) Suppose | A | = 8. If A is not a sum of two linear words, then we say that A is a quadratic word . (iii) Suppose | A | = 9. If A is neither a sum of three linear words nor a sum of a linear and a quadratic words, then we say that A is a cubic word . By C �→ w ( C ), we obtain the following bijections: { lines splitting in ( X G , L G ) } ∼ { linear words in C G } , = { non-singular conic curves splitting in ( X G , L G ) } ∼ = { quadratic words in C G } . By E �→ w ( E t ) = γ − 1 (Bs( E )), we obtain the bijection { regular pencils of cubic curves splitting in ( X G , L G ) } ∼ { cubic words in C G } . = 13

§ 4. The Case of Artin Invariant 2 We start from a code C ⊂ 2 P such that • P = (1 , 1 , . . . , 1) ∈ C , and • | w | ∈ { 0 , 5 , 8 , 9 , 12 , 13 , 16 , 21 } for any w ∈ C , and construct the stratum M [ C ] . For simplicity, we assume that C is generated by P and words of weight 5 and 8. We denote by G C the space of all injective maps → P 2 γ : P ֒ with the following properties: (i) γ ( P ) = Z ( dG ) for some G ∈ U (that is, γ ∈ G ), (ii) for a subset A ⊂ P of weight 5, γ ( A ) is collinear if and only if A ∈ C , (iii) for a subset A ⊂ P of weight 8, γ ( A ) is on a non- singular conic curve if and only if A ∈ C and A is not a sum of words of weight 5 in C . 14