Tropical Discriminants Eva Maria Feichtner feichtne@math.ethz.ch Department of Mathematics, ETH Zurich E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.1/25
Outline 1. A -Discriminants 2. Tropical Geometry 3. Tropical A -Discriminants 4. The Newton Polytope of ∆ A 5. Regular Subdivisions and ∆ -Equivalence of Triangulations joint work/project with Alicia Dickenstein and Bernd Sturmfels E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.2/25
A -Discriminants 1. [Gelfand, Kapranov, Zelevinsky 1992] ∈ Z d × n , rk A = d , � � A = a 1 · · · a n (1 , . . . , 1) ∈ row span A Q A = conv { a 1 , . . . , a n } polytope in R d , dim Q A = d − 1 X A = V ( � x u − x v | u, v ∈ N n with Au = Av � ) projective toric variety A = cl { ξ ∈ ( CP n − 1 ) ∗ | H ξ tangent to X A at a regular point } X ∗ dual variety If codim X ∗ A = 1 , X ∗ A = V (∆ A ) , where ∆ A is a unique irreducible polynomial, the A -discriminant. E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.3/25
A -Discriminants: Classical Examples 1. Discriminant of a quadratic polynomial in 1 variable f ( x ) = a 2 x 2 + a 1 x + a 0 , a 2 � = 0 ∆ f = a 2 ⇐ ⇒ 1 − 4 a 2 a 0 = 0 f has a double root � � 1 1 1 ∆ f = ∆ A ∈ Z [ a 0 , a 1 , a 2 ] for A = 0 1 2 2. Discriminant of a degree n polynomial in 1 variable f ( x ) = � n i =0 a i x i , a n � = 0 f has a double root ⇐ ⇒ ∆ f = 0 � � 1 1 . . . 1 ∆ f = ∆ A ∈ Z [ a 0 , . . . , a n ] for A = 0 1 . . . n E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.4/25
A -Discriminants: Classical Examples 2. Resultant of two polynomials in 1 variable n m � � a i x i , b i x i , a n � = 0 , b m � = 0 , f ( x ) = g ( x ) = i =0 i =0 ⇐ ⇒ f and g have a common root Res( f, g ) = 0 Res( f, g ) = ∆ A ∈ Z [ a 0 , . . . , a n , b 0 , . . . , b m ] for 1 1 . . . 1 0 0 . . . 0 A = 0 0 . . . 0 1 1 . . . 1 0 1 . . . n 0 1 . . . m Res( f, g ) = determinant of the Sylvester matrix E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.5/25
A -Discriminants: Classical Examples 3. Discriminant of a deg 2 homogeneous polynomial in 3 variables 2 1 1 0 0 0 2 a 1 a 2 a 3 A = ∆ A = det 0 1 0 2 1 0 a 2 2 a 4 a 5 0 0 1 0 1 2 a 3 a 5 2 a 6 4. Discriminant of a deg 3 homogeneous polynomial in 3 variables 1 1 1 1 1 1 1 1 1 1 A = 0 0 0 0 1 1 1 2 2 3 0 1 2 3 0 1 2 0 1 0 deg ∆ A = 12 , 2040 terms E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.6/25
A -Discriminants Call A defective if codim X ∗ A > 1 . The dual variety X ∗ A is also of interest in the defective case. Goal: Derive information on ∆ A , resp. X ∗ A , for instance degree and extreme monomials of ∆ A dimension, degree and Chow form of X ∗ A directly from A , without any reference to defining equations. Ansatz: Study the tropicalization of X ∗ A ! E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.7/25
2. Tropical Geometry ( R ∪ {∞} , ⊕ , ⊗ ) , x ⊕ y := min { x, y } , x ⊗ y := x + y tropical semi-ring complex projective τ − → polyhedral fans varieties Y ⊆ CP n − 1 irreducible variety, dim Y = r I Y ⊆ C [ x 1 , . . . , x n ] defining prime ideal τ ( Y ) = { w ∈ R n | in w ( I Y ) does not contain a monomial } tropicalization of Y τ ( Y ) is a pure r -dimensional polyhedral fan in R n , respectively TP n − 1 = R n / R (1 , 1 , . . . , 1) . E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.8/25
Examples of Tropicalized Varieties 1. Y hypersurface in CP n − 1 f ∈ C [ x 1 , . . . , x n ] irreducible polynomial defining Y New( f ) Newton polytope, N New( f ) its normal fan τ ( Y ) = codim 1 -skeleton of N New( f ) Proof: { w ∈ R n | in w ( f ) is not a monomial } τ ( Y ) = { w ∈ R n | dim � � > 0 } = New(in w ( f )) { w ∈ R n | dim � � > 0 } = w -maximal face of New( f ) � = σ σ ∈N New( f ) codim σ> 0 E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.9/25
Examples of Tropicalized Varieties 2. Y = X A toric variety, A ∈ Z d × n τ ( Y ) = row span A Proof: � x u − x v | u, v ∈ N n with Au = Av � I X A = { w ∈ R n | in w ( f ) is not a monomial for any f ∈ I X A } τ ( Y ) = { w ∈ R n | wu = wv whenever Au = Av } = = row span A 3. Y = V linear, resp. projective subspace τ ( Y ) = B ( M ( V )) Bergman fan of the matroid associated with V E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.10/25
Digression: Bergman Fans of Matroids M connected matroid on { 1 , . . . , n } , rk M = r M w = { σ ∈ M | σ basis with maximal w -cost } for w ∈ R n B ( M ) = { w ∈ R n | M w is loop-free } Bergman fan w ∈ R n � � � � � w 2 B ( M ) = B ( M ) ∩ w i = 0 , i = 1 � � Bergman complex B ( M ) is a (rk M − 1) -dimensional subfan of N P ( M ) , where P ( M ) is the matroid polytope of M . E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.11/25
Examples of Bergman Fans M = M ( K 4 ) r = 3 , n = 6 1 2 3 125 6 146 1 3 6 5 5 236 4 345 2 4 K 4 B ( M ( K 4 )) M = M ( K 4 \ e ) r = 3 , n = 5 2 4 3 125 1 3 5 345 1 2 4 K 4 \ e B ( M ( K 4 \ e )) E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.12/25
Digression: Nested Set Fans of Matroids M connected matroid on { 1 , . . . , n } , rk M = r L M lattice of flats G ⊆ ( L M ) > ˆ 0 building set if for any X ∈ L M and max G ≤ X = { G 1 , . . . , G k } , there exists an isomorphism k � [ˆ [ˆ φ X : 0 , G i ] − → 0 , X ] . i =1 G min : irreducibles, dense edges, connected flats L M \ { ˆ G max : 0 } S ⊆ G nested set if for any pairwise incomparable X 1 , . . . , X t ∈ S , t ≥ 2 , � X i �∈ G . N ( G ) abstract simplicial complex of nested sets N ( G ) realization as a simplicial fan in R n E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.13/25
Examples of Nested Set Fans M = M ( K 4 ) r = 3 , n = 6 1 1 3 3 125 125 146 146 6 6 5 5 1 2 3 4 5 6 236 236 345 345 2 4 4 2 L M N ( G min) N ( G max) M = M ( K 4 \ e ) r = 3 , n = 5 4 3 4 125 125 3 1 2 5 3 4 1 2 345 1 2 345 N ( G max ) N ( G min ) L ′ M E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.14/25
Bergman Fans versus Nested Set Fans Proposition: [F. & Sturmfels ’04; F. & M¨ uller ’03] B ( M ) is subdivided by N ( G ) for any building set G in L M . N ( G ) is subdivided by N ( G ′ ) for any building sets G ⊆ G ′ in L M . Back to tropical geometry: V a linear subspace in C n , M the associated matroid, G any building set in L M τ ( V ) = supp B ( M ) = supp N ( G ) E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.15/25
Tropical A -Discriminants 3. ∈ Z d × n , rk A = d , � � a 1 · · · a n (1 , . . . , 1) ∈ row span A A = Horn uniformization of A -discriminants: [Kapranov ’91] The dual variety X ∗ A is the closure of the image of the morphism P (ker A ) × ( C ∗ ) d / C ∗ ( CP n − 1 ) ∗ − → ϕ A : ( u 1 t a 1 : u 2 t a 2 : · · · : u n t a n ) . �− → ( u, t ) Tropical Horn uniformization: TP n − 1 B (ker A ) × R d − → τ ( ϕ A ) : �− → ( w, v ) w + vA im τ ( ϕ A ) = B (ker A ) + row span A Horn fan E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.16/25
Tropical A -Discriminants Theorem: [DFS ’05] τ ( X ∗ A ) = B (ker A ) + row span A Example: 2 1 1 0 0 0 A = 0 1 0 2 1 0 0 0 1 0 1 2 5 245 5 4 2 1 6 4 356 1 123 6 τ ( X ∗ A ) B (ker A ) 3 3 2 E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.17/25
Tropicalizing Monomials in Linear Forms U V f : C m → C r → C s − − U ∈ C r × m linear map, V ∈ Z s × r monomial map r � ( u k 1 x 1 + · · · + u km x m ) v ik f i ( x 1 , . . . , x m ) = k =1 Y UV := closure of im f Examples: r = s , V = I r : Y UV = im U linear space m = r , U = I m : Y UV = X V T toric variety Theorem: [DFS ’05] τ ( Y UV ) = V ◦ τ (im U ) = V ◦ B ( M (im U )) E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.18/25
Tropicalizing Monomials in Linear Forms Retrieving the tropical discriminant: Set m = r , r = n + d , s = d , B a Gale dual of A , � � B 0 � A T � U = and V = I n . 0 I d UV = X ∗ Then, Y P A , and τ ( X ∗ A ) = B (ker A ) + row span A . E.M. Feichtner: Tropical Discriminants; Algebraic and Geometric Combinatorics, Anogia, August 21, 2005. – p.19/25
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