Résume | The secret goal of this talk is to explain a little of the magic of log line bundles. The vehicle for this will be a story about double ramification cycles for roots of a line bundle. Given a family of curves C/S and a line bundle L on C, the double ramification cycle DR(L) is a class on S measuring the set of points in S over which L is trivial (or more precisely, where L is trivial as a log line bundle). The formal goal of this talk is to describe a lift to the universal r-th root of L. More precisely, for a positive integer r we define the stack of r-th roots of L, which is a finite flat cover of S of degree r^{2g}. It carries a universal r-th root of L (as a log line bundle), and the locus where this r-th root is (logarithmically) trivial defines a lift of DR(L) to the stack of r-th roots. Pixton's formula for DR(L) admits a fairly straightforward lift to this setting. |