Using Lie algebras to parametrize certain types of algebraic - PowerPoint PPT Presentation

Using Lie algebras to parametrize certain types of algebraic varieties II Willem A. de Graaf, University of Trento, Italy Janka P´ ılnikov´ a, RICAM Linz / RISC Linz, Austria Josef Schicho, RICAM Linz, Austria Lie Algebras, their Classification and Applications University of Trento 25 - 27 July 2005 July 25, 2005 Page 1

Outline (i) Surfaces in algebraic geometry, parametrizing surfaces (ii) The Lie algebra of a surface. (iii) Using the method to parametrize non-trivial cases of Del Pezzo surfaces of degree 8: – “non-split” case: the unit sphere as a twist of P 1 × P 1 – “non-semisimple” case: the blowup of P 2 July 25, 2005 Page 2

Surfaces in projective space The n -dimensional projective space: P n = { ( t 0 : · · · : t n ) | ( t 0 : · · · : t n ) � = (0 : · · · : 0) } . Let f 1 , . . . , f k ∈ Q [ x 0 , . . . , x n ] are forms over Q . Projective variety S is the set of solutions to f 1 , . . . , f k , i.e. all points in P n such that f 1 ( a 0 , . . . , a n ) = · · · = f k ( a 0 , . . . , a n ) = 0 . Let S be a surface: dim( S ) = 2 . Finding a rational parametrization over Q means finding all rational solutions to the system. We need a map ϕ : P 2 ( or P 1 × P 1 ) → S ⊂ P n subject to the following: July 25, 2005 Page 3

Parametrization of a surface S ⊂ P n a surface. ϕ : P 2 ( or P 1 × P 1 ) → P n is a rational parametrization of S over Q , if (i) ϕ = ( p 0 : · · · : p n ) and p 0 , . . . p n ∈ Q [ t 0 , t 1 , t 2 ] are forms of the same degree, (ii) ϕ ( p ) ∈ S for all p from the domain of ϕ , (iii) the image of ϕ is a dense subset of S . Preprocessing: By embeddings of S we reduce to few basic cases: – tubular surfaces – some trivial cases – Del Pezzo surfaces of degree 5, 6, 7, 8, 9. July 25, 2005 Page 4

The Lie algebra of a surface For a given surface S ⊂ P n , G ( S, Q ) := { g ∈ GL n +1 ( Q ) | ∀ p ∈ S gp ∈ S } . L ( S, Q ) := Lie( G ( S, Q )) – the Lie algebra of S . How to compute L ( S, Q ) ? If S ⊂ P 3 is given by a quadratic form f ( x ) = x T Ax over Q : S = { p = ( x 0 : x 1 : x 2 : x 3 ) T | p T Ap = 0 } , then G ( S, Q ) = { g ∈ GL 4 ( Q ) | g T Ag = λA } and L ( S, Q ) = { X ∈ gl 4 ( Q ) | X T A + AX = λA } . If S ⊂ P n is given by a set of quadratic forms f 1 , . . . , f k over Q , f i ( x ) = x T A i x : S = { p = ( x 0 : · · · : x n ) T | p T A i p = 0 ∀ i } , then G ( S, Q ) = { g ∈ GL n +1 ( Q ) | g T A i g ∈ � A 1 , . . . A k � Q ∀ i } and L ( S, Q ) = { X ∈ gl n +1 ( Q ) | X T A i + A i X ∈ � A 1 , . . . A k � Q ∀ i } . July 25, 2005 Page 5

Parametrizing twists of P 1 × P 1 The canonical surface S 0 : x 1 x 2 = x 0 x 3 is parametrized by P 1 × P 1 : ϕ 0 : ( s 0 : s 1 ; t 0 : t 1 ) �→ ( s 0 t 0 : s 0 t 1 : s 1 t 0 : s 1 t 1 ) = ( x 0 : x 1 : x 2 : x 3 ) . G ( S 0 , Q ) = { g ∈ GL 4 ( Q ) | ∀ p ∈ S 0 gp ∈ S 0 } . The Lie algebra L ( S 0 , Q ) = Lie( G ( S 0 , Q )) is isomorphic to sl 2 ( Q ) ⊕ sl 2 ( Q ) ⊕ Q . L ( S 0 , Q ) ⊂ gl 4 ( Q ) . The module V 0 of L ( S 0 , Q ) is 4-dimensional irreducible sl 2 ⊕ sl 2 -module with the heighest weight (1 , 1) . July 25, 2005 Page 6

Short review of the basic method Example: S : x 2 0 − x 2 1 − x 2 2 + x 2 3 is projectivelly equivalent to S 0 over Q : - the Lie algebra L ( S, Q ) ∼ = sl 2 ( Q ) ⊕ sl 2 ( Q ) ⊕ Q . - the module W of L ( S, Q ) is 4-dimensional irredcible sl 2 ⊕ sl 2 -module with the heighest weight (1 , 1) . The isomorphism ψ : V 0 → W : e 0 �→ v 3 + v 0 , e 1 �→ v 3 − v 0 , e 2 �→ v 2 + v 1 , e 3 �→ v 2 − v 1 is unique, up to multiplication by scalars. Therefore ψ is also the projective equivalence of S 0 and S ψ : ( x 0 : x 1 : x 2 : x 3 ) �→ ( x 3 + x 0 : x 3 − x 0 : x 2 + x 1 : x 2 − x 1 ) and gives us a parametrization of S : ϕ = ψ ◦ ϕ 0 : ( x 0 : x 1 : x 2 : x 3 ) = ( s 1 t 1 + s 0 t 0 : s 1 t 1 − s 0 t 0 : s 1 t 0 + s 0 t 1 : s 1 t 0 − s 0 t 1 ) . July 25, 2005 Page 7

Sphere as a twist of P 1 × P 1 The unit sphere S : x 2 1 + x 2 2 + x 2 3 = x 2 0 is not isomorphic to S 0 ( x 1 x 2 = x 0 x 3 ) over Q : L ( S, Q ) = L 0 ( S, Q ) ⊕ I 4 , where L 0 ( S, Q ) is 6-dimensional simple Lie algebra which is a twist of sl 2 ⊕ sl 2 . But still S has a rational parametrization. We find a splitting field E of L 0 ( S, Q ) as the centroid of the algebra: Let E be the centralizer of ad( L 0 ( S, Q )) in gl ( L 0 ( S, Q )) . Then E = Q ( i ) and L 0 ( S, E ) ∼ = sl 2 ( E ) ⊕ sl 2 ( E ) . The corresponding module becomes sl 2 ⊕ sl 2 -module over E with maximal weight (1 , 1) . We get a parametrization of S over E : ψ : S 0 → S : ( x 0 : x 1 : x 2 : x 3 ) = ( − s 0 t 1 + s 1 t 0 : s 0 t 0 − s 1 t 1 : is 0 t 0 + is 1 t 1 : s 0 t 1 + s 1 t 0 ) . July 25, 2005 Page 8

Sphere as a twist of P 1 × P 1 – continued F 1 , F 2 – the two families of lines on the surface. F 1 is a 1-dimensional family of lines over E : ∀ ( s : t ) ∈ P 1 ( E ) l ( s : t ) ∈ F 1 . For the centroid E we have [ E : Q ] = 2 . Let σ be the nontrivial automorphism of E over Q . If l ∈ F 1 then σ ( l ) ∈ F 2 . Therefore l ∩ σ ( l ) = { p } . p is fixed under σ , hence p is a rational point and ( s : t ) �→ l ( s : t ) ∩ σ ( l ( s : t ) ) is a map P 1 ( E ) → S ( Q ) . The projective line P 1 ( E ) can be parametrized by the projective plane P 2 ( Q ) : ( a : b : c ) �→ ( a + ib : c ) . This leads to a rational parametrization of the sphere ( a : b : c ) �→ ( c 2 + a 2 + b 2 : 2 ac : − 2 bc : c 2 − a 2 − b 2 ) with a, b, c, ∈ Q . July 25, 2005 Page 9

Parametrizing blowups of P 2 The canonical blowup S 0 ⊂ P 8 is parametrized ( s : t : u ) �→ ( s 2 t : s 2 u : st 2 : stu : su 2 : t 3 : t 2 u : tu 2 : u 3 ) . Let S ⊂ P 8 be projectively equivalent to S 0 over Q . The Lie algebras of S 0 and S decompose as a sum of sl 2 ( Q ) and a 3-dimensional radical R : ϕ 0 : sl 2 ( Q ) + R → L ( S 0 , Q ) , ϕ : sl 2 ( Q ) + R → L ( S, Q ) . As sl 2 -modules: V ( ϕ 0 ) = W 2 ( ϕ 0 ) ⊕ W 3 ( ϕ 0 ) ⊕ W 4 ( ϕ 0 ) , V ( ϕ ) = W 2 ( ϕ ) ⊕ W 3 ( ϕ ) ⊕ W 4 ( ϕ ) with dim( W i ( ϕ 0 )) = dim( W i ( ϕ )) = i . July 25, 2005 Page 10

Parametrizing blowups of P 2 – continued (1) Any isomorphism ψ : V ( ϕ 0 ) → V ( ϕ ) maps W i ( ϕ 0 ) to W i ( ϕ ) , i = 2 , 3 , 4 . P ( W 2 ( ϕ 0 )) ( P ( W 2 ( ϕ )) ) is the exceptional line of S 0 ( S ). One can use geometric methods to parametrize S . (2) Consider V ( ϕ 0 ) as an ( sl 2 + R ) -module: Elements of the radical carry W i ( ϕ 0 ) to W i − 1 ( ϕ 0 ) , i = 3 , 4 , so V ( ϕ 0 ) is irreducible. The same with V ( ϕ ) . The isomorphism ψ : V ( ϕ 0 ) → V ( ϕ ) as ( sl 2 + R ) -modules is unique up to multplication by scalars. Therefore it is also an isomorphism of S 0 and S and hence a parametrization of S . July 25, 2005 Page 11