An embedding theorem for smooth projective toric varieties (Q1119977)

It is known that a compact projective toric variety \(X_{\Delta}\) can be constructed by means of a convex polytope \(\Delta\) obtained as intersection of translated cones. On the other hand a general embedding of \(X_{\Delta}\) in a projective space \({\mathbb{P}}^ k\) is not a complete intersection. The author proves that any n-dimensional smooth complete projective toric variety \(X_{\Delta}\) can be embedded into a projective space \({\mathbb{P}}^ k\) as zero locus of finitely many homogeneous monomial equations. Moreover he finds out, both for k and for the number q of such equations, a bound related to the number of vertices of \(\Delta\).