Smooth Compactifications of the Abel-Jacobi Section

TL;DR

This paper introduces smooth, modular blowups of the moduli space of stable curves that facilitate the calculation of the logarithmic double ramification cycle, building on recent compactification techniques.

Contribution

It demonstrates that certain compactifications of the Abel-Jacobi section are smooth and modular, enabling new methods for computing the double ramification cycle.

Findings

Spaces $ar{M}_{g,n}^ heta$ lie inside the space $ extbf{Div}$ and are smooth after pullback.

Provides explicit smooth and modular blowups $ ilde{M}_{g,n}^ heta$ of the moduli space.

Enables calculation of the logarithmic double ramification cycle using multiple methods.

Abstract

For $\theta$ a small generic universal stability condition of degree $0$ and $A$ a vector of integers adding up to $k(2g-2)$, the spaces $\overline{M}_{g,n}^\theta$ resolving the Abel-Jacobi section to the compactified Jacobian Pic^\theta constructed in the work of Abreu-Pacini and Holmes-Molcho-Pandharipande-Pixton-Schmitt are observed to lie inside the space $\textbf{Div}$ of Marcus and Wise, and their pullback to the rubber space of loc. cit to be smooth. This provides smooth and modular blowups $\widetilde{M}_{g,n}^\theta$ of the moduli space of stable curves on which the logarithmic double ramification cycle can be calculated by several methods, old and novel.