Math 274 : Tropical Geometry , Homework 5

Proof. [Solution due to Claudiu Raicu] Consider two polynomials f, g ∈ C[X±] such that the spherical graphs they determine intersect transversely at the points where the vectors vi intersect the sphere. This implies that T (f) and T (g) intersect transversely ([1]) at each point and T (f)∩T (g) = Σ. Using the Transverse Intersection Lemma (lemma 3.2, corollary 3.3 of [1]) we get that Σ = T (f, g), so it suffices to prove that we can choose f, g so that (f, g) is prime. We can then take I = (f, g) ∩ C[X]. We now show how to choose f, g: clearly T (f), T (g) depend only on the monomials in f, g, so we can fix these sets of monomials (and denote them A, B). Abusing notation we write f = ∑ α∈A aαX , g = ∑ β∈B bβX β where aα, bβ are indeterminates. Since the generic fiber of the natural map f : Spec C[X±, aα, bβ ] → Spec C[aα, bβ ] is integral and we are working over an algebraically closed field it follows that there is an open set where the fiber of f is integral, and since closed points are dense in schemes of finite type over fields we can find a closed point in that open set. This will give the desired f, g.