Radical Sylvester-Gallai Theorem for Cubics

Abstract

We prove that any cubic radical Sylvester-Gallai configuration is constant dimensional. This solves a conjecture of Gupta in degree 3 and generalizes the result from Shpilka, who proved that quadratic radical Sylvester-Gallai configurations are constant dimensional. To prove our Sylvester-Gallai theorem, we develop several new tools combining techniques from algebraic geometry and elimination theory. Among our technical contributions, we prove a structure theorem characterizing non-radical ideals generated by two cubic forms, generalizing previous structure theorems for intersections of two quadrics. Moreover, building upon the groundbreaking work Ananyan and Hochster, we introduce the notion of wide Ananyan-Hochster algebras and show that these algebras allow us to transfer the local conditions of Sylvester-Gallai configurations into global conditions.