Comments on the ELSV compactification of Hurwitz stacks

TL;DR

This paper analyzes the ELSV compactification of Hurwitz stacks, connecting it with admissible covers and providing a foundation for combinatorial interpretation of boundary points.

Contribution

It establishes a link between the ELSV compactification and the Harris-Mumford admissible covers, clarifying boundary structures and paving the way for combinatorial analysis.

Findings

Boundary of ELSV compactification is a contraction of admissible covers boundary

Provides foundational understanding for combinatorial interpretation

Connects different compactification approaches in Hurwitz theory

Abstract

We revisit Ekedahl, Lando, Shapiro and Vainshtein's compactification of the stack of simply ramified covers of the projective line except for a fixed ramification profile above infinity. In particular we draw a connection with the Harris and Mumford stack of admissible covers showing that the boundary of the ELSV compactification appears as a contraction of the boundary of the stack of admissible covers. This lays the needed foundations for a combinatoral interpretation of boundary points of the ELSV compactification