Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals

TL;DR

This paper investigates symbolic powers of bi-homogeneous ideals of points in P^1 x P^1, extending known results to this setting, and provides a classification of point configurations with minimal fattening effects.

Contribution

It extends results on points fattening to bi-homogeneous ideals in P^1 x P^1 and proves a Chudnovsky-type theorem for this setting.

Findings

Proved a Chudnovsky-type theorem for bi-homogeneous ideals.

Classified configurations of points with minimal or no fattening effect.

Extended algebraic and geometric applications to arbitrary surfaces.

Abstract

We study symbolic powers of bi-homogeneous ideals of points in the Cartesian product of two projective lines and extend to this setting results on the effect of points fattening obtained by Bocci, Chiantini and Dumnicki, Szemberg, Tutaj-Gasi\'nska. We prove a Chudnovsky-type theorem for bi-homogeneous ideals and apply it to classification of configurations of points with minimal or no fattening effect. We hope that the ideas developed in this project will find further algebraic and geometric applications e.g. to study similar problems on arbitrary surfaces.