Abstract | Abstract Consider the ideal I ⊆ K [ x , y , z ] corresponding to points p_1 , … , p_r of P^2 . We study the symbolic generic initial system gin ( I ( m ) ) m of such an ideal and its behaviour as m gets large. In particular, we describe the limiting shape of this system explicitly when p 1 , … , p r are generic, assuming that the Uniform SHGH Conjecture holds for r ≥ 9 . The symbolic generic initial system and its limiting shape reflect information about the asymptotic behaviour of Hilbert functions of fat point ideals. |