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We show that two ostensibly different versions of the asymptotic resurgenceintroduced by Guardo, Harbourne and Van Tuyl in 2013 are the same. We also showthat the resurgence and asymptotic resurgence attain their maximal valuessimultaneously, if at all, which we apply to a conjecture of Grifo. For radicalideals of points, we show that the resurgence and asymptotic resurgence attaintheir minimal values simultaneously. In addition, we introduce an integralclosure version of the resurgence and relate it to the other versions of theresurgence. In closing we provide various examples and raise some relatedquestions, and we finish with some remarks about computing the resurgence.

Extreme values of the resurgence for homogeneous ideals in polynomial rings