# Algebraic Geometry 3 Curves Jaobians by Parshin Shafarevich

By Parshin Shafarevich

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CONVEXITY, FINITENESS, AND DUALITY 45 Proof: Let G be a face of fine monoid Q. The natural map Q → Q/G induces a bijection between the facets of Q containing F and the facets of Q/F . Thus, replacing Q by Q/F , we reduce to the case in which Q is sharp and F = 0. We must show in this case that if q ∈ Q belongs to every facet of Q, then q = 0. The complement of a facet F is a prime ideal p of height one, and q ∈ F if and only if νp (q) = 0. Since the set of all such νp generates the cone CQ (H(Q)), it follows that h(q) = 0 for all h ∈ H(Q).

19 (Gordon’s lemma) Let L be a finitely generated abelian group, let V := Q ⊗ L, and let C ⊆ Q ⊗ L be a finitely generated Q-cone. Then CL := L ×V C ∼ = L ×VR CR is a finitely generated monoid. 2). Let us first treat the case in which L is free, so that it may be identified with its image in V . Let S be a finite set of generators for C, which we may as well assume contained in L. Let S ⊆ VR be the set of all linear combinations of elements of S with coefficients in the interval [0, 1]. The map [0, 1]S → VR sending {as : s ∈ S} to as s is continuous and maps surjectively to S ; hence S is compact.

Let µ be the gp composition M → Mp → N, then µ−1 (N+ ) = p, and νp := µgp is an epimorphism such that νp−1 (N+ ) ∩ M = p. Suppose that ν: M gp → Z is an epimorphism such that ν −1 (N+ ) ∩ M = p. Then ν −1 (0) ∩ M is the face gp F := M \ p, and ν factors through Mp ∼ = Z. Since ν is an epimorphism, this last map is an isomorphism, and ν = ±νp . In fact the sign must be + since ν −1 (N+ ) = p. If q and p are elements of M , νp (p − q) = νp (p) − νp (q). Thus if q ∈ M \ p, νp (q) = 0 and νp (p − q) ≥ 0.