Quadratic generation of ideals defining nonsigular toric 3-folds

Abstract

Let $X$ be a projective line bundle over a nonsingular toric surface which is a blowup the projective plane along at most 4 invariant points, or a blowup the product of the projective lines along at most 4 invariant points. Let $L$ be an ample line bundle on $X$. Then $L$ defines a projectively normal embedding to big projective space. We show the ideal of this embedded $X$ is generated by elements of degree two.