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Consider a one-parameter family (f_t, a_t), where f_t is a polynomial and a_t is a marked point, and t ranges over points of an algebraic curve C. We discuss a result (due originally to DeMarco in the more general setting of rational maps) that classifies such families based on the preperiodic locus in C, i.e. {t such that a_t is preperiodic under f_t}. We’ll follow the exposition in the textbook by Favre and Gauthier (https://arxiv.org/pdf/2004.13801.pdf.)