Computing toric degenerations of flag varieties Sara Lamboglia - PowerPoint PPT Presentation

Computing toric degenerations of flag varieties Sara Lamboglia University of Warwick with Lara Bossinger, Kalina Mincheva and Fatemeh Mohammadi (arXiv 1702.05505 )

Compute G r¨ obner toric degenerations of F ℓ 4 and F ℓ 5 Compare them with the degenerations obtained using representation theory techniques ( Littelman(1998),Berenstein-Zelevinsky(2001), Caldero(2002),Alexeev-Brion (2005)).

Why toric degenerations ? Toric varieties give a powerful dictionary which translates combinatorial properties to algebraic and geometric properties. P P 1 × P 1 P P 3

Why toric degenerations? = ⇒ Extend this dictionary to a larger class of varieties. 1 for Use a toric degeneration , i.e a flat family ϕ : F → A which the fibre over 0 is a toric variety and all the other fibres are isomorphic to the variety F ℓ n .

Why flag varieties? Let k be any field. Definition The set of all complete flags V : { 0 } = V 0 � V 1 � · · · � V n − 1 � V n = k n in k n is denoted by F ℓ n and it has an algebraic variety structure.

Why flag varieties? Let k be any field. Definition The set of all complete flags V : { 0 } = V 0 � V 1 � · · · � V n − 1 � V n = k n in k n is denoted by F ℓ n and it has an algebraic variety structure.

Why flag varieties? Let k be any field. Definition The set of all complete flags V : { 0 } = V 0 � V 1 � · · · � V n − 1 � V n = k n in k n is denoted by F ℓ n and it has an algebraic variety structure.

Why flag varieties? Let k be any field. Definition The set of all complete flags V : { 0 } = V 0 � V 1 � · · · � V n − 1 � V n = k n in k n is denoted by F ℓ n and it has an algebraic variety structure. F ℓ n can be embedded in Gr ( 1 , k n ) × · · · × Gr ( n − 1 , k n ) . It can also be seen as SL n / B .

Why flag varieties? Let k be any field. Definition The set of all complete flags V : { 0 } = V 0 � V 1 � · · · � V n − 1 � V n = k n in k n is denoted by F ℓ n and it has an algebraic variety structure. F ℓ n can be embedded in Gr ( 1 , k n ) × · · · × Gr ( n − 1 , k n ) . It can also be seen as SL n / B . = ⇒ Flag varieties are a good toy model because of their additional structures.

Pl¨ ucker embedding F ℓ n := {V : { 0 } = V 0 � V 1 � · · · � V n − 1 � V n = k n } F ℓ n ⊂ Gr ( 1 , k n ) × · · · × Gr ( n − 1 , k n ) Using Pl¨ ucker embeddings F ℓ n becomes a subvariety of P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and it has defining ideal I n ⊂ k [ p J : ∅ � = J � { 1 , . . . , n } ] .

Example: F ℓ 3 Let n = 3 then F ℓ 3 = { ( ℓ, H ) ∈ Gr ( 1 , k 3 ) × Gr ( 2 , k 3 ) : ℓ ⊂ H } . = P 2 × P 2 . It is a subvariety of Gr ( 1 , k 3 ) × Gr ( 2 , k 3 ) ∼ It is defined in k [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] by the ideal I 3 = � p 3 p 12 − p 2 p 13 + p 1 p 23 � .

Example: F ℓ 3 Let n = 3 then F ℓ 3 = { ( ℓ, H ) ∈ Gr ( 1 , k 3 ) × Gr ( 2 , k 3 ) : ℓ ⊂ H } . = P 2 × P 2 . It is a subvariety of Gr ( 1 , k 3 ) × Gr ( 2 , k 3 ) ∼ It is defined in k [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] by the ideal I 3 = � p 3 p 12 − p 2 p 13 + p 1 p 23 � .

Example: F ℓ 3 Let n = 3 then F ℓ 3 = { ( ℓ, H ) ∈ Gr ( 1 , k 3 ) × Gr ( 2 , k 3 ) : ℓ ⊂ H } . = P 2 × P 2 . It is a subvariety of Gr ( 1 , k 3 ) × Gr ( 2 , k 3 ) ∼ It is defined in k [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] by the ideal I 3 = � p 3 p 12 − p 2 p 13 + p 1 p 23 � .

Toric degenerations 1 for which the fibre We are looking for a flat family ϕ : F → A over 0 is a toric variety and all the other fibres are isomorphic to the variety F ℓ n . After the embedding we have F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and F ℓ n = V ( I n ) .

Toric degenerations 1 for which the fibre We are looking for a flat family ϕ : F → A over 0 is a toric variety and all the other fibres are isomorphic to the variety F ℓ n . After the embedding we have F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and F ℓ n = V ( I n ) . 1 ) − 1 × · · · × P ( n n − 1 ) − 1 are defined by toric Toric varieties inside P ( n ideals, i.e. binomial and prime.

Toric degenerations 1 for which the fibre We are looking for a flat family ϕ : F → A over 0 is a toric variety and all the other fibres are isomorphic to the variety F ℓ n . After the embedding we have F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and F ℓ n = V ( I n ) . 1 ) − 1 × · · · × P ( n n − 1 ) − 1 are defined by toric Toric varieties inside P ( n ideals, i.e. binomial and prime. 1 such that the fibre over 0 = ⇒ We need a flat family ϕ : F → A is defined by a toric ideal, i.e. binomial and prime and the general fibre is isomorphic to V ( I n ) .

Toric degenerations 1 for which the fibre We are looking for a flat family ϕ : F → A over 0 is a toric variety and all the other fibres are isomorphic to the variety F ℓ n . After the embedding we have F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and F ℓ n = V ( I n ) . 1 ) − 1 × · · · × P ( n n − 1 ) − 1 are defined by toric Toric varieties inside P ( n ideals, i.e. binomial and prime. 1 such that the fibre over 0 = ⇒ We need a flat family ϕ : F → A is defined by a toric ideal, i.e. binomial and prime and the general fibre is isomorphic to V ( I n ) . = ⇒ Consider Gr¨ obner degenerations.

Gr¨ obner toric degenerations Definition Let f = � a u x u with u ∈ Z n be a polynomial in k [ x 1 , . . . , x n ] . For each w ∈ R n we define its initial form to be � a u x u . in w ( f ) = w · u is minimal

Gr¨ obner toric degenerations Definition Let f = � a u x u with u ∈ Z n be a polynomial in k [ x 1 , . . . , x n ] . For each w ∈ R n we define its initial form to be � a u x u . in w ( f ) = w · u is minimal Example: generator of I 3 Consider k [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] and the polynomial f = p 3 p 12 − p 2 p 13 + p 1 p 23 = = p ( 0 , 0 , 1 , 1 , 0 , 0 ) − p ( 0 , 1 , 0 , 0 , 1 , 0 ) + p ( 1 , 0 , 0 , 0 , 0 , 1 )

Gr¨ obner toric degenerations Definition Let f = � a u x u with u ∈ Z n be a polynomial in k [ x 1 , . . . , x n ] . For each w ∈ R n we define its initial form to be � a u x u . in w ( f ) = w · u is minimal Example: generator of I 3 Consider k [ p 1 , p 2 , p 3 , p 12 , p 13 , p 23 ] and the polynomial f = p 3 p 12 − p 2 p 13 + p 1 p 23 = = p ( 0 , 0 , 1 , 1 , 0 , 0 ) − p ( 0 , 1 , 0 , 0 , 1 , 0 ) + p ( 1 , 0 , 0 , 0 , 0 , 1 ) then in ( 1 , 0 , 0 , 0 , 0 , 0 ) ( f ) = p 3 p 12 − p 2 p 13

Definition If I is an ideal in S , then its initial ideal with respect to w is in w ( I ) = � in w ( f ) : f ∈ I � . 1 for which the fibre over 0 There exists a flat family ϕ : F → A is isomorphic to V ( in w ( I )) and all the other fibres are isomorphic to the variety V ( I ) . This is called a Gr¨ obner degeneration of V ( I ) .

Example: F ℓ 3 For F ℓ 3 the defining ideal is I 3 = � p 3 p 12 − p 2 p 13 + p 1 p 23 � . If w = ( 1 , 0 , 0 , 0 , 0 , 0 ) then in w ( I 3 ) = � p 3 p 12 − p 2 p 13 � which is prime and binomial hence it defines a toric variety. The flat family defining this toric degeneration is given by I t = � p 3 p 12 − p 2 p 13 + tp 1 p 23 �

Example: F ℓ 3 For F ℓ 3 the defining ideal is I 3 = � p 3 p 12 − p 2 p 13 + p 1 p 23 � . If w = ( 1 , 0 , 0 , 0 , 0 , 0 ) then in w ( I 3 ) = � p 3 p 12 − p 2 p 13 � which is prime and binomial hence it defines a toric variety. The flat family defining this toric degeneration is given by I t = � p 3 p 12 − p 2 p 13 + tp 1 p 23 �

Given F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and let I n be the defining ideal, i.e. F ℓ n = V ( I n ) . Problem Find embedded (possibly not normal) toric degenerations of V ( I n ) .

Given F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and let I n be the defining ideal, i.e. F ℓ n = V ( I n ) . Problem Find embedded (possibly not normal) toric degenerations of V ( I n ) . Using Gr¨ obner degenerations the problem translates in

Given F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and let I n be the defining ideal, i.e. F ℓ n = V ( I n ) . Problem Find embedded (possibly not normal) toric degenerations of V ( I n ) . Using Gr¨ obner degenerations the problem translates in Algebraic reformulation Find toric initial ideals of I n .

Given F ℓ n ⊂ P ( n 1 ) − 1 × · · · × P ( n n − 1 ) − 1 and let I n be the defining ideal, i.e. F ℓ n = V ( I n ) . Problem Find embedded (possibly not normal) toric degenerations of V ( I n ) . Using Gr¨ obner degenerations the problem translates in Algebraic reformulation Find toric initial ideals of I n . Consider the tropicalization of X .

Tropicalization Let I ⊂ k [ x 1 , ..., x n ] and X = V ( I ) . Definition The tropicalization trop ( X ) of X is defined to be { w ∈ R n : in w ( I ) does not contain monomials }

Tropicalization Let I ⊂ k [ x 1 , ..., x n ] and X = V ( I ) . Definition The tropicalization trop ( X ) of X is defined to be { w ∈ R n : in w ( I ) does not contain monomials } The tropical variety trop ( X ) has a fan structure such that in w ( I ) = in w ′ ( I ) for all w ′ , w in the relative interior of a cone C ∈ trop ( X ) . Each cone C corresponds to a different initial ideal.