Bounding the Multigraded Regularity of Powers of Ideals

Building on a result of Swanson, Cutkosky--Herzog--Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space: given an ideal I in a polynomial ring over a field, there exist integers d and e such that for n sufficiently large the regularity of I^n satisfies reg(I^n) = dn + e.

Through a medley of examples we will see why asking the same question about I an ideal in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) inside Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way we will also see some surprising behavior for multigraded regularity of modules.