A Gröbner basis for an ideal under an elimination order reflects much of the geometric structure of the variety defined by the ideal. We discuss how this relationship can be used in decomposing polynomial systems (with or without parameters) and in primary decomposition of ideals. As an application, we show how this technique can be used in designing deterministic algorithms for factoring univariate polynomials over finite fields, which aims to reduce the factoring problem to a combinatorial one. The talk is based on joint work with Genhua Guan, Raymond Heindl, Jand-Woo Park, Virginia Rodrigues, and Jeff Stroomer.