On the ampleness of invertible sheaves in complete projective toric varieties (Q1174922)

Given a rational partial polyhedral decomposition \(\Delta\) of \(\mathbb{R}^ n\), i.e. a complete fan in \(\mathbb{R}^ n\), one can associate to it a toric variety \(X\). It is well-known that \(X\) is projective iff there exists a \(\Delta\)-linear support function \(h\) strictly upper convex with respect to \(\Delta\), i.e. \(h\) is a real valued function on \(\mathbb{R}^ n\) which is \(\mathbb{Z}\)-valued on lattice points, linear on cones of \(\Delta\) and satisfies certain convexity condition. To such a function one can associate an ample Cartier divisor \(D_ h\), i.e. \(kD_ h\) gives a projective embedding of \(X\) for sufficiently large \(k\). Let \(k_ 0\) be the minimum of such \(k\). It is known that \(k_ 0=1\) if \(X\) is nonsingular or if \(X\) is complete and \(n=2\). The authors prove that \(k_ 0\leq n-1\). To prove this result, the condition for very ampleness of \(kD_ h\) is expressed in combinatorial language as a condition on the integral convex polyhedron (lattice polytope) canonically associated to \(h\). The proof is completely combinatorial.