Positive Charts of Toric Varieties

This is a real manifold with corners, isomorphic to a d-simplex. None of the standard affine charts y i ̸ = 0 contains P d ≥0 . A convenient alternative is the chart U = P d \V (y 0 +• • •+y d ) ⊃ P d ≥0 . The restriction of the quotient representation π : C d+1 \ {0} → P d to the hyperplane

uniquely picks a set of homogeneous coordinates π -1 (p) ∩ Y for each p ∈ U . In particular, the semi-algebraic subset of Y given by y i ≥ 0 is identified with P d ≥0 . Note that Y is the image of

Hence, it admits a positive rational parametrization whose Newton polytope is isomorphic to P d ≥0 as a real manifold with corners. Here “positive” means that all numerators and denominators have nonnegative coefficients, and the Newton polytope of a rational parametrization is the Minkowski sum of the Newton polytopes of all these polynomials. That is, (Newt(r i ) + Newt(s i ))

where r i and s i are coprime. Our paper extends these constructions to any smooth projective toric variety, providing a positive rational parametrization for its nonnegative part.

Let Σ be a smooth rational fan in R d . The smooth toric variety X Σ is the GIT quotient

where Σ(1) is the set of rays of Σ, Z(Σ) is a union of coordinate subspaces and the quotient is by the action of the torus G = Hom Z (Cl(X Σ ), C * ) [Cox95]. A point y = (y ρ ) ρ∈Σ(1) ∈ C Σ(1) \ Z(Σ) gives homogeneous coordinates or Cox coordinates for its image under the map

This is the quotient morphism associated to (2). For instance, if X Σ = P d , then Z(Σ) = {0}, G = C * and π -1 (p) ⊂ C d+1 \ {0} is the C * -orbit of homogeneous coordinate choices for p.

The nonnegative part of X Σ consists of all points with nonnegative Cox coordinates:

≥0 ̸ = ∅}.

Let Σ = Σ P be the normal fan of a smooth polytope P . We have that (X Σ ) ≥0 is isomorphic to P as a real manifold with corners [Ful93, §4.2]. One of our main results is a constructive proof of the following theorem, which generalizes the above discussion for X Σ = P d . Notice that point 3 in Theorem 1.1 implies that (X Σ ) ≥0 ⊆ U . An interpretation of point 1 is that Y picks a unique set of Cox coordinates π -1 (p) ∩ Y for each point p ∈ U . In Section 3, we show how to compute a positive chart for the toric variety of any smooth polytope P . In particular, we construct the parametrization from point 2 explicitly, and we shall derive defining equations for Y ∩ (C * ) Σ(1) . The following will serve as our running example.

Example 1.3. We consider the normal fan Σ of a pentagon in R 2 as in Figure 1, and we explicitly describe one of its positive charts. The base locus Z(Σ) is given by the ideal ⟨y 1 y 2 y 3 , y 1 y 2 y 5 , y 3 y 4 y 5 , y 1 y 4 y 5 , y 2 y 3 y 4 ⟩ and the group G ∼ = (C * ) 3 acts on C 5 \ Z(Σ) as follows (λ, µ, ν) • (y 1 , y 2 , y 3 , y 4 , y 5 ) = (λ y 1 , ν λ y 2 , λµ ν y 3 , ν µ y 4 , µ y 5 ), λ, ν, µ ∈ C * .

The affine variety Y ⊂ C 5 \ Z(Σ) is a complete intersection cut out by three equations

(1, 0) It is obtained as the image of the following positive rational parametrization φ : C 2 Y :

The Newton polytope of this map is a pentagon whose normal fan is Σ:

The open set U is the complement of a union of three irreducible curves in X Σ . These are the closures in X Σ of the curves in (C * ) 2 defined by 1 + t 1 = 0, 1 + t 2 = 0, 1 + t 2 + t 1 t 2 = 0. ⋄ In Section 5 we associate a natural moment map to a positive chart Y . That is, we construct explicit morphisms µ Y,s : Y → C Σ(1) depending on parameters s which identify Y ≥0 with a d-dimensional polytope P (s) whose normal fan is Σ P (s) = Σ. We describe the fibers µ -1 Y,s (x) as the critical points of a multivalued monomial function ρ∈Σ(1) y xρ ρ on Y ∩ (C * ) Σ(1) .

While not phrased in the language of homogeneous coordinates on toric varieties, the recipe followed in Section 3 to construct Y and U first appeared in the physics literature [AHL21, Sections 9.5 and 10]. The variety Y plays the role of a binary geometry [AHHLT23], in the sense that it recovers several examples from physics for specific choices of Σ (see Section 4). In particular, our paper identifies the dihedral coordinates (or u-coordinates) on the moduli space M 0,n from [Bro12, Section 2] as Cox coordinates on the toric variety associated with a smooth realization of the associahedron. Example 1.3 appears in [AHL21, AHHLT23,Bro12], and Y is shown to be a partial compactification of M 0,5 in [Bro12].

The recent paper [AHFP + 25] presents a large family of examples of Y -varieties arising in a representation-theoretic context. There, the fan Σ is the g-vector fan of a finite representation type C-algebra, and such an algebra serves as the starting point of the construction of Y . In contrast, our point of departure is the fan Σ itself. We prove Theorem 1.1 using only standard techniques from toric geometry; no representation theory enters the argument. It would be interesting to further expl