Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in PN. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P3.