Introduction and the Main Theorem An example Sketch of Proof Implementation Almost-Toric Hypersurfaces Bo Lin University of California, Berkeley April 18th, 2015 STAGS 2015, Brown University arXiv:1410.0776 Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Abstract An almost-toric hypersurface is parametrized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit equation. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Toric Part of Z A, f Let A be a n × ( n + 2) full-rank matrix with integer entries. � � A = a 0 a 1 . . . a n +1 . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Toric Part of Z A, f Let A be a n × ( n + 2) full-rank matrix with integer entries. � � A = a 0 a 1 . . . a n +1 . If the entries of each vector a i have the same sum d , then t a 0 , t a 1 , . . . , t a n +1 are Laurent monomials in n variables t 1 , . . . , t n of the same degree d , where t a i = t a 1 ,i t a 2 ,i . . . t a n,i . n 1 2 Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Coefficient Part of Z A, f We fix K as an algebraically closed field. Let f 0 , f 1 , . . . , f n +1 ∈ K [ x ] be univariate polynomials in another variable x . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Almost-Toric Hypersurface Z A, f Definition An n -dimensional almost-toric hypersurface Z A, f is a codimension 1 hypersurface that is the Zariski closure of following parametrization: { ( t a 0 f 0 ( x ) : . . . : t a n +1 f n +1 ( x )) ∈ P n +1 | t ∈ ( K ∗ ) n , x ∈ K } . Remark We need some mild hypothesis of A, f to guarantee that Z A, f is a codimension 1 hypersurface. We will see the hypothesis in our main theorem. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Newton Polytope Suppose Z A, f is an almost-toric hypersurface, then its ideal is principal. The Newton polytope of Z A, f is defined as the Newton polytope of the generator (up to a scalar multiple) of this principal ideal. This generator, denoted as p ( u 0 , . . . , u n +1 ) , is called the implicit equation of Z A, f . Proposition The Newton polytope of an almost-toric hypersurface Z A, f is at most 2 -dimensional in R n +2 . Its vertices are non-negative lattice points. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Newton Polygon Proof. If we substitute all u i by t a i f i ( x ) in p , we get another polynomial in variables t 1 , . . . , t n , x . By definition this is the zero polynomial. After this substitution, each term in p becomes the product of a monomial in variables t 1 , t 2 , . . . , t n and a polynomial in x . Because p is the generator of a principle ideal, all such monomials in variables t 1 , t 2 , . . . , t n are the same, which gives n independent linear equations on the vertices of Newt( Z A, f ) and we conclude that Newt( Z A, f ) is at most 2 -dimensional. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Main Theorem Theorem Suppose Z A, f is defined as before. It has a Pl¨ ucker matrix P A for its toric part and a valuation matrix V f for its coefficients. (a) if rank( P A · V f ) = 0 then Z A, f is not a hypersurface; (b) if rank( P A · V f ) = 1 then Z A, f is a toric hypersurface; (c) if rank( P A · V f ) = 2 then Z A, f is a hypersurface but not toric. The directed edges of the Newton polygon of Z A, f are the nonzero column vectors of P A · V f . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Main Theorem Theorem Suppose Z A, f is defined as before. It has a Pl¨ ucker matrix P A for its toric part and a valuation matrix V f for its coefficients. (a) if rank( P A · V f ) = 0 then Z A, f is not a hypersurface; (b) if rank( P A · V f ) = 1 then Z A, f is a toric hypersurface; (c) if rank( P A · V f ) = 2 then Z A, f is a hypersurface but not toric. The directed edges of the Newton polygon of Z A, f are the nonzero column vectors of P A · V f . Remark The sum of each row in P A · V f is zero. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Pl¨ ucker Matrices The Pl¨ ucker matrix P A of A is a ( n + 2) × ( n + 2) square matrix whose row vectors span the kernel of A . The i, j -th entry of P A is δ ( − 1) i + j det( A [ i,j ] ) , 1 i < j ; p i,j = − p j,i , i > j ; 0 , i = j. where A [ i,j ] is the submatrix obtained from A by deleting the i -th and j -th columns of A and δ is the greatest common divisor of all det( A [ i,j ] ) . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Pl¨ ucker Matrices Example: � 3 � 2 1 0 A = . 0 1 2 3 Then δ = 3 and 0 − 1 2 − 1 1 0 − 3 2 P A = . − 2 3 0 − 1 1 − 2 1 0 Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Pl¨ ucker Matrices Example: � 3 � 2 1 0 A = . 0 1 2 3 Then δ = 3 and 0 − 1 2 − 1 1 0 − 3 2 P A = . − 2 3 0 − 1 1 − 2 1 0 Proposition P A is skew-symmetric. The rank of P A is 2 . The entries in each row and column of P A sum to 0 . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Valuation Matrices The valuation matrix of Z A, f is defined from the polynomials f 0 , f 1 , . . . , f n +1 . Suppose g 1 , g 2 , . . . , g m are all irreducible factors of � n +1 i =0 f i . Then we define vectors u j = deg( g j ) · ( ord g j f 0 , ord g j f 1 , . . . , ord g j f n +1 ) ∈ N n +2 for 1 ≤ j ≤ m . Now we need to simplify the set of these vectors. We combine the pairwise linearly dependent vectors, because they correspond to the same edge of the polygon. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Valuation Matrices - continued Now let S = { u 1 , u 2 , . . . , u m } . If two vectors in S are linearly dependent, then we delete them and add their sum to the set. We repeat this procedure. After finite steps, we end up with another set without pairwise linearly dependent vectors. Finally we get S ′ = { v 1 , v 2 , . . . , v l } . Definition The valuation matrix of Z A, f is � j =1 v j ) T � ( − � l v T v T v T V f = . . . . 1 2 l The last vector represents the valuation at ∞ . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Valuation Matrices - continued Now let S = { u 1 , u 2 , . . . , u m } . If two vectors in S are linearly dependent, then we delete them and add their sum to the set. We repeat this procedure. After finite steps, we end up with another set without pairwise linearly dependent vectors. Finally we get S ′ = { v 1 , v 2 , . . . , v l } . Definition The valuation matrix of Z A, f is � j =1 v j ) T � ( − � l v T v T v T V f = . . . . 1 2 l The last vector represents the valuation at ∞ . Proposition The sum of each row in V f is zero. Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Valuation Matrices Example: n = 2 , f 0 = x − 1 , f 1 = ( x − 1) 2 ( x +1) , f 2 = ( x +1) x 3 , f 3 = ( x − 2) x. Then the irreducible factors are x − 1 , x, x + 1 , x − 2 . 1 x − 1 corresponds to (1 , 2 , 0 , 0) 2 x corresponds to (0 , 0 , 3 , 1) 3 x + 1 corresponds to (0 , 1 , 1 , 0) 4 x − 2 corresponds to (0 , 0 , 0 , 1) These vectors are pairwise linearly independent, so the valuation matrix is 1 0 0 0 − 1 2 0 1 0 − 3 V f = . 0 3 1 0 − 4 0 1 0 1 − 2 Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation 3 -dimensional hypersurface Z A, f Let Z A, f admit the following parametrization over C : ( t 2 1 ( x 2 +1) : t 1 t 2 x 3 ( x − 1) : t 1 t 3 x ( x +1) : t 2 2 ( x − 2)( x 2 +1) : t 2 3 ( x − 1) 2 ( x +1)) . Bo Lin Almost-Toric Hypersurfaces
Introduction and the Main Theorem An example Sketch of Proof Implementation Pl¨ ucker Matrix of Z A, f In this example 2 1 1 0 0 , d = δ = 2 . A = 0 1 0 2 0 0 0 1 0 2 Then 0 − 2 2 1 − 1 2 0 − 4 0 2 P A = − 2 4 0 − 2 0 . − 1 0 2 0 − 1 1 − 2 0 1 0 Bo Lin Almost-Toric Hypersurfaces
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