Finite $0$-dimensional multiprojective schemes and their ideals

Abstract

We study finite 0-dimensional schemes in product of multiprojective space and their ideals. In particular, we describe the set of generators of the ideal defining a 0-dimensional scheme in the case P 1 × • • • × P 1 and in the case P n 1 × • • • × P n k with n i ∈ {1, 2} for all i. We also check very ampleness for zero-dimensional schemes of general points in the multiprojective spaces.