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We study symbolic powers of bi-homogeneous ideals of points in the Cartesianproduct of two projective lines and extend to this setting results on theeffect of points fattening obtained by Bocci, Chiantini and Dumnicki, Szemberg,Tutaj-Gasi\'nska. We prove a Chudnovsky-type theorem for bi-homogeneous idealsand apply it to classification of configurations of points with minimal or nofattening effect. We hope that the ideas developed in this project will findfurther algebraic and geometric applications e.g. to study similar problems onarbitrary surfaces.

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