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Brief Guide to Some of the Literature on F-singularities

Many of the tools of higher dimensional complex birational algebraic geometry—including singularities of pairs, multiplier ideals, and log canonical thresholds— have ”characteristic p” analogs arising from ideas in tight closure theory. Tight closure, introduced by Hochster and Huneke in [17], is a closure operation performed on ideals in commutative rings of prime characteristic, and has an independent trajectory as an active branch of commutative algebra. Huneke’s book [19] gives a nice introduction from the algebraic perspective. Here we review some of the connections with birational geometry that have developed since the survey [30] appeared. Basic references for the birational geometry terms used here are [24], and [21] or [20].