Finite $0$-dimensional multiprojective schemes and their ideals

TL;DR

This paper investigates the structure of ideals defining finite zero-dimensional schemes in products of multiprojective spaces, providing explicit generators and examining very ampleness for schemes of points.

Contribution

It characterizes generators of ideals for zero-dimensional schemes in specific multiprojective products and analyzes their very ampleness properties.

Findings

Explicit generators for ideals in P^1×...×P^1 cases.

Generators for schemes in P^{n_1}×...×P^{n_k} with n_i in {1,2}.

Verification of very ampleness for schemes of general points.

Abstract

We study finite $0$-dimensional schemes in product of multiprojective spaces and their ideals. In particular, we describe the set of generators of the ideal defining a $0$-dimensional scheme in the case $\mathbb P^{1}\times\cdots \times\mathbb P^{1}$ and in the case $\mathbb P^{n_1}\times \cdots \times \mathbb P^{n_k}$ with $n_i\in \{1,2\}$ for all $i$. We also check very ampleness for zero-dimensional schemes of general points in the multiprojective spaces.