Grbner-basis-free proofs of freeness of hyperplane arrangements - PowerPoint PPT Presentation

Gröbner-basis-free proofs of freeness of hyperplane arrangements Mohamed Barakat University of Kaiserslautern Algebra, Algorithms, and Algebraic Analysis Abdij Rolduc, 6. September 2013. joint work with T. Abe, M. Cuntz, T. Hoge, and H. Terao Mohamed Barakat Gröbner-basis-free proofs of freeness

Freeness of divisors • k a field and S = k [ x 1 , . . . , x ℓ ] • A ℓ = Spec S = S ℓ is the free S -module of • Der = Der A ℓ = � ℓ i =1 S∂ x i ∼ polynomial derivations or polynomial vector fields . Definition Let D = { f = 0 } ⊂ A ℓ be a divisor: • The submodule of logarithmic derivations is defined as � � d f ≤ Der ∼ = S ℓ . Der( − log D ) = ker Der − → S → S/ � f � These are all infinitesimal external symmetries of D in the sense of S OPHUS L IE . • D is called free if Der( − log D ) is a free S -module. One can compute Der( − log D ) and decide its freeness using G RÖBNER bases over S . Mohamed Barakat Gröbner-basis-free proofs of freeness

The N EIL parabola gap> LoadPackage( "Sheaves" );; gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q * "x,y"; Q[x,y] gap> AssignGeneratorVariables( R ); #I Assigned the global variables [ x, y ] gap> f := y^2-x^3; -x^3+y^2 gap> D := Divisor( f ); <A divisor on <An affine space A^2 over Q>> gap> DerD := DerMinusLog( D ); <A torsion-free left submodule given by 2 generators> gap> IsFree( DerD ); true gap> DerD; <A free left submodule of rank 2 on free generators> gap> Display( DerD ); 2*x,3*y, 2*y,3*x^2 A left submodule generated by the 2 rows of the above matrix Mohamed Barakat Gröbner-basis-free proofs of freeness

The N EIL parabola (continued) Mohamed Barakat Gröbner-basis-free proofs of freeness

Hyperplane arrangements • A = { H 1 , . . . , H n } a hyperplane arrangement in A ℓ • α H ∈ S a degree one defining equation of H ∈ A • ∪A := { � H ∈A α H = 0 } is the associated divisor Definition A is called free if D ( A ) := Der( − log ∪A ) is free. • If A is central then D ( A ) is a graded S -module. • If A is central and free then D ( A ) = � ℓ i =1 S ( − d i ) . • { d 1 ≤ · · · ≤ d ℓ } is called the multiset of exponents of A . Root systems provide an interesting class of examples. Mohamed Barakat Gröbner-basis-free proofs of freeness

Root systems Let V be an ℓ -dimensional real vector space. A reflection along α ∈ V is an R -linear map s satisfying • s ( α ) = − α , • H s := Fix( s ) is a hyperplane. Definition A root system Φ ⊂ V \ { 0 } is a finite generating system of V such that 1 for each α ∈ Φ there exists a reflection s α with s α ( R ) = R (and hence exactly one), 2 for each α, β ∈ Φ : s α ( β ) ∈ β + Z α . • Φ is called reduced if � α � = { α, − α } . • Φ is called irreducible if it cannot be decomposed as a direct sum of two root subsystems Φ 1 ⊕ Φ 2 ⊂ V 1 ⊕ V 2 = V . Mohamed Barakat Gröbner-basis-free proofs of freeness

Classification of root systems Example Let ( e 1 , . . . , e ℓ ) denote the standard basis of R ℓ . The following subsets of V are reduced irreducible root systems • A ℓ − 1 := { e i − e j } i � = j ⊂ R ℓ − 1 • D ℓ := A ℓ − 1 ∪ {± ( e i + e j ) } i � = j ⊂ R ℓ • B ℓ := D ℓ ∪ {± e i } ⊂ R ℓ • C ℓ := D ℓ ∪ {± 2 e i } ⊂ R ℓ Theorem The above example is an exhaustive list except for 5 sporadic root systems E 6 , E 7 , E 8 , F 4 , G 2 . From now on all root systems will be assumed reduced and irreducible. Mohamed Barakat Gröbner-basis-free proofs of freeness

Simple system Definition • Fix a set Φ + of positive roots , i.e., Φ + ˙ ∪ − Φ + = Φ . • The corresponding simple system ∆ := Φ + \ { α + β | α, β ∈ Φ + } ⊂ Φ + is a basis of V . • ∆ is the only antichain in Φ + with ℓ roots. • The simple system ∆ := { α 1 , . . . , α ℓ } turns Φ + into a poset ℓ � α ≥ β : ⇐ ⇒ α − β ∈ Z ≥ 0 α i i =1 • An (order) ideal I in a poset is a subset closed under passage to smaller elements. Mohamed Barakat Gröbner-basis-free proofs of freeness

C OXETER arrangements are inductively free Set H α = H s α = Fix( s α ) for α ∈ Φ + . • For an ideal I ⊂ Φ + we call A ( I ) := { H α | α ∈ I } the ideal subarrangement of I . • A (Φ + ) is called a W EYL arrangement . Using invariant theory one can show that Theorem ([O RLIK , T ERAO ]) W EYL arrangements are free. The exponents can be described combinatorially. Theorem (–, C UNTZ ) Crystallographic & C OXETER arrangements are inductively free. Our proof is heavily computer assisted. Mohamed Barakat Gröbner-basis-free proofs of freeness

Ideal subarrangements are free Theorem (A BE , –, C UNTZ , H OGE , T ERAO ) Any ideal subarrangement A ( I ) ⊂ A (Φ + ) is free. The exponents can be described combinatorially. Proof. Make an induction on the height �� � � ht λ α α = λ α α ∈ ∆ α ∈ ∆ and use the following multiple addition theorem to add the roots of largest height simultaneously . Mohamed Barakat Gröbner-basis-free proofs of freeness

The multiple addition theorem (MAT) • A ′ be a free central arrangement • H 1 , . . . , H q new central hyperplanes (i.e., not in A ′ ) with q -codimensional and �⊂ ∪A ′ X := H 1 ∩ · · · ∩ H q j := A ′ ∩ H j and assume that • Define the restrictions A ′′ |A ′ | − |A ′′ ( ⋆ ) j | = d ∀ j = 1 , . . . , q, where d is the highest exponent of A Theorem (loc. cit.) Then A := A ′ ∪ { H 1 , . . . , H q } is free. The exponents of A are expressible in terms of those of A ′ . Mohamed Barakat Gröbner-basis-free proofs of freeness

Idea of the proof. • Let θ 1 , . . . , θ ℓ be a free basis of D ( A ′ ) . • Use ( ⋆ ) to show that θ 1 , . . . , θ ℓ − p ∈ D ( A ) , where p is the multiplicity of the highest exponent d . • For z ∈ X �⊂ ∪A ′ p vectors � �� � T V,z = � θ 1 , . . . , θ ℓ − p � z ⊕ � θ ℓ − p +1 , . . . , θ ℓ � z � �� � � �� � ⊂ T X,z ⇒ ∼ = T V,z /T X,z = � �� � dim= q • Hence p ≥ q and we may assume that the last q vectors ( θ ℓ − q +1 , . . . , θ ℓ ) form a basis of T V,z /T X,z . • Again using ( ⋆ ) one can show that ( θ 1 , . . . , θ ℓ − q , α H q θ ℓ − q +1 , . . . , α H 1 θ ℓ ) is a free basis of D ( A ) . Mohamed Barakat Gröbner-basis-free proofs of freeness

Thank you Mohamed Barakat Gröbner-basis-free proofs of freeness

Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, and Hiroaki Terao, The freeness of ideal subarrangements of weyl arrangements , submitted ( arXiv:1304.8033 ), 2013. Mohamed Barakat and Michael Cuntz, Coxeter and crystallographic arrangements are inductively free , Adv. Math. 229 (2012), no. 1, 691–709, ( arXiv:1011.4228 ). MR 2854188 (2012i:20048) Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR MR1217488 (94e:52014) Mohamed Barakat Gröbner-basis-free proofs of freeness