Abstract:

We derive an explicit combinatorial formula for the double ramification cycle of type (1,-1) on the moduli space of stable genus g curves with two marked points. The formula is given as a sum over certain strata on the moduli space of curves, indexed by so-called extremal trees. We present two different proofs of the formula: one using a local equivariant method and the other using blow up and piecewise polynomial techniques. From this main result we obtain similarly-formatted variant formulas, as well as tautological relations in higher codimension. We also study the compact type double ramification cycle of type (2,-2), establish its connection with hyperelliptic admissible cover loci, and give some examples. These works link Gromov-Witten type cycles with admissible cover cycles on the moduli space of curves.