This talk will be a discussion of some interesting and novel subgroups of the mapping class group that arise via algebro-geometric constructions. Our talk will focus on the special case of how the theory of plane algebraic curves (essentially just polynomials in two variables!) interacts with the mapping class group in subtle ways. The motivating question can be formulated simply as, ``which mapping classes (of a surface of genus g) arise as one-parameter families of polynomials in two variables?’’ Perhaps surprisingly, the answer turns out to be ``either none at all, or else virtually all of them”. No familiarity with algebraic geometry will be assumed.