Lines in the tropics Maria Angelica Cueto Department of Mathematics - PowerPoint PPT Presentation

Lines in the tropics Maria Angelica Cueto Department of Mathematics The Ohio State University Blackwell-Tapia Conference 2018 - ICERM Based on joint works in preparation with Anand Deopurkar (Australia) and Hannah Markwig (Germany) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 1 / 28

Tropical Mathematics SLOGAN 1: Tropical Geometry is Algebraic Geometry over the tropical semifield ( R , ⊕ , ⊙ ). SLOGAN 2: Tropical varieties are combinatorial shadows of algebraic varieties (over valued fields.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 2 / 28

SLOGAN 1: Trop. Geometry is Alg. Geometry over R tr :=( R , ⊕ , ⊙ ). • R = R ∪ {−∞} , a ⊕ b = max { a , b } , a ⊙ b = a + b (E.g.: 3 ⊕ 5 = 5, 3 ⊙ 5 = 8, −∞ ⊕ 3 = 3, 0 ⊙ 3 = 3.) n → R cont., convex, affine PL with Z -slopes Polys in R tr [ X 1 , . . . , X n ] ≡ R a α ⊙ X ⊙ α 1 � ⊙ · · · ⊙ X ⊙ α n F ( X ) = 1 n α ∈ N n 0 (finite) = max α { a α + α 1 X 1 + . . . + α n X n } • Tropical hypersurface: V tr ( F ) = corner locus of F . • Examples: M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 3 / 28

SLOGAN 2: Trop. vars are comb. shadows of alg. vars via valuations . • Def.: A valuation on a field K is a map val: K � { 0 } → R satisfying: (1) val( xy ) = val( x ) + val( y ), (2) val( x + y ) � min { val( x ) , val( y ) } (and = if val( x ) � = val( y )) Extend val to K via val(0) = + ∞ . Examples: • Trivial valuations val( x ) = 0 for all x � = 0. • K = C ( ( t ) ) with t -valuation (val(2 t − 5 +3 t − 1 / 2 + . . . )= − 5) . • K = Q p with p -adic valuation. • We tropicalize polynomials in K [ x 1 , . . . , x n ] using ( − val on K , ⊕ and ⊙ ): a α x α � trop ( f )( X )= max � f ( x )= {− val( a α )+ α 1 X 1 + . . . + α n X n } α ∈ Supp( f ) α n (max is at two α ’s) • Def. 1: Trop( V ( f )) = Corner locus of trop( f ) in R In general: I defining ideal � Trop( I ) = � Trop( V ( f )). f ∈ I M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 4 / 28

SLOGAN 2 (cont.) : Trop. vars are comb. shadows of alg. vars via valns . Fix K = K with non-trivial valn. (e.g. K = C ( ( t ) )). → Y Σ = toric variety with dense torus ( K ∗ ) n . Fix a closed embedding ι : X ֒ = ( K ∗ ) n , K n or P n . Examples: Y Σ n or TP n := R n +1 � { ( −∞ ,..., ∞ ) } R n , R Trop Y Σ = ≃ ∆ n ( n -simplex) . R · 1 Def. 2: Trop X =cl . { ( − val( p 1 ) , . . . , − val( p n )): ( p 1 , . . . , p n ) ∈ X }⊂ Trop Y Σ Fundamental Thm. of Trop. Geom.: Both definitions agree. Structure Thm.: Trop( X ) is a polyhedral complex of dimension dim( X ) (pure if X is irreducible, balanced if multiplicities on top-dim. cells.) ISSUE: Definition of Trop( X ) is coordinate dependent! (Q: Best choices?) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 5 / 28

Examples: Lines in the tropics over K = C ( ( t ) ) • Example 0: The line K � Trop( K ) = R • Example 1: The line 1 + x + y = 0 in the plane K 2 . Def. 1: f = 1 + x + y � trop( f )( X , Y ) = max { 0 , X , Y } 2 → K 2 Def. 2: ι : K ֒ ι ( x ) = ( x , − 1 − x ) � ( − val( x ) , − val(1 + x )) in R • Example 2: Trop. Lines in TP 2 • Example 3: Trop. Lines in TP 3 M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 6 / 28

Tropical plane curves = metric graphs in R 2 = dual to Newton subdivisions Example: f ( x , y ) = t + x + y + x y + 2 t x 2 + (3 t + t 2 ) y 2 in C ( ( t ) )[ x , y ] trop( f )( X , Y )= max {− 1 , X , Y , X + Y , − 1 + 2 X , − 1 + 2 Y } 0. Take a polynomial f in K [ x , y ] with K non-trivially valued field. 1. Build the Newton Polytope of f : NP ( f ) :=conv(( i , j ) in supp( f )). 2. Place each point ( i , j ) from NP ( f ) at height − val(coeff( x i y j )) in R 3 . M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 7 / 28

Tropical plane curves = metric graphs in R 2 = dual to Newton subdivisions Example: f ( x , y ) = t + x + y + x y + 2 t x 2 + (3 t + t 2 ) y 2 in C ( ( t ) )[ x , y ] trop( f )( X , Y )= max {− 1 , X , Y , X + Y , − 1 + 2 X , − 1 + 2 Y } 0. Take a polynomial f in K [ x , y ] with K non-trivially valued field. 1. Build the Newton Polytope of f : NP ( f ) :=conv(( i , j ) in supp( f )). 2. Place each point ( i , j ) from NP ( f ) at height − val(coeff( x i y j )) in R 3 . 3. Take upper hull and project to R 2 . We get a subdivision of NP ( f ). M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 8 / 28

Tropical plane curves = metric graphs in R 2 = dual to Newton subdivisions Example: f ( x , y ) = t + x + y + x y + 2 t x 2 + (3 t + t 2 ) y 2 in C ( ( t ) )[ x , y ] trop( f )( X , Y )= max {− 1 , X , Y , X + Y , − 1 + 2 X , − 1 + 2 Y } 0. Take a polynomial f in K [ x , y ] with K non-trivially valued field. 1. Build the Newton Polytope of f : NP ( f ) :=conv(( i , j ) in supp( f )). 2. Place each point ( i , j ) from NP ( f ) at height − val(coeff( x i y j )) in R 3 . 3. Take upper hull and project to R 2 . We get a subdivision of NP ( f ). 4. Trop( V ( f )) = dual graph to this subdivision. Comes with a metric. M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 9 / 28

Basic Facts about general tropical plane curves: (1) Interpolation for general pts in R 2 holds tropically (Mikhalkin’s Corresp.) (unique line through 2 gen. points, unique conic through 5 gen. points,. . . ) (2) General trop. curves intersect properly and as expected (Trop. B´ ezout.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 10 / 28

Basic Facts about general tropical plane curves: (1) Interpolation for general pts in R 2 holds tropically (Mikhalkin’s Corresp.) (unique line through 2 gen. points, unique conic through 5 gen. points,. . . ) (2) General trop. curves intersect properly and as expected (Trop. B´ ezout.) Non-general case: Replace usual intersection with stable intersection. C 1 ∩ st C 2 := ε → (0 , 0) C 1 ∩ ( C 2 + ε ) . lim M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 11 / 28

Today’s focus: 2 classical results in Algebraic Geometry ucker (1834): A sm. quartic curve in P 2 Pl¨ C has exactly 28 bitangent lines. (0,4,8,16 or 28 real bitangents, depending on topology of the real curve.) Salmon: 28 real, 24 totally real. Trott: 28 totally real bitangents. Cayley-Salmon (1849): Any smooth algebraic cubic surface in P 3 C contains exactly 27 distinct lines. Figure: Clebsch cubic surface ISSUE: Both results fail tropically! But we can fix it. M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 12 / 28

28 bitangent lines to sm. plane quartics over K = C ( ( t ) ). ucker (1834): A sm. quartic curve in P 2 Pl¨ K has exactly 28 bitangent lines. (0,4,8,16 or 28 real bitangents, depending on topology of the real curve.) Question: What happens tropically? Baker-Len-Morrison-Pflueger-Ren (2015): Every tropical smooth quartic in R 2 has 7 bitangent classes. Len-Markwig (2017): Generically, each class lifts to 4 classical bitangents. Len-Jensen (2017): Each class always lifts to 4 classical bitangents. Question: What is a tropical bitangent line? Need tangencies at 2 points. Len-Markwig: 5 local tangencies (up to S 3 -symmetry.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 13 / 28

28 bitangent lines to sm. plane quartics over K = C ( ( t ) ). Theorem: There are 28 classical bitangents to sm. plane quartics over K but 7 tropical bitangent classes to their tropicalizations. Question: Combinatorial proof? Trop. sm. quartic=dual to unimodular triangulation of ∆ 2 of side length 4. � duality gives a genus 3 planar metric graph. u w w w Possible cases: w x u y y v y z u u y v z u z v y z x z x x w v v x M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 14 / 28

28 bitangent lines to sm. plane quartics over K = C ( ( t ) ). Theorem: There are 28 classical bitangents to sm. plane quartics over K but 7 tropical bitangent classes to their tropicalizations. Question: Combinatorial proof? Trop. sm. quartic=dual to unimodular triangulation of ∆ 2 of side length 4. � duality gives a genus 3 planar metric graph. u w w w Possible cases: x u y w y y v z u u z u y v z v y z x z x x [BLMPR ’15] w v v x Brodsky-Joswig-Morrison-Sturmfels (2015): Newton subdivisions give linear restrictions on the lengths u , v , w , x , y , z of the edges. M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 15 / 28

28 classical bitangents vs. 7 tropical bitangent classes. Local tangencies: (up to symm.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 16 / 28

28 classical bitangents vs. 7 tropical bitangent classes. Local tangencies: (up to symm.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 17 / 28

28 classical bitangents vs. 7 tropical bitangent classes. Local tangencies: (up to symm.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 18 / 28

28 classical bitangents vs. 7 tropical bitangent classes. Local tangencies: (up to symm.) M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 19 / 28

28 classical bitangents vs. 7 tropical bitangent classes. M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 20 / 28

28 classical bitangents vs. 7 tropical bitangent classes. M.A. Cueto (Ohio State) Lines in the tropics November 10th 2018 21 / 28