The double ramification cycle and the theta divisor

Abstract:

We compute the classes of universal theta divisors of degrees zero and $g-1$ over the Deligne-Mumford compactification ${\overline {\mathcal {M}}_{g,n}}$ of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller.

We also obtain a formula for the class in $CH^{g}({\mathcal {M}_{g,n}^{ct}})$ (moduli of stable curves of compact type) of the double ramification cycle, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain.

Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over all of $\overline {\mathcal {M}}_{g,n}$. We used our extended result in another paper to study the partial compactification of the double ramification cycle.