We revisit Ekedahl, Lando, Shapiro and Vainshtein’s compactification of the stack of Hurwitz covers. By drawing a connection with the Harris and Mumford stack of admissible covers we give a new geometric interpretation of boundary points of the ELSV compactification. As a byproduct we establish that this compactification holds for any algebraically closed field of sufficiently high characteristic.

We find here another (not of the CROC-type) compactification of the Kontsevich spaces K(m,n). This compactification is a Stasheff-type compactification, in particular, it is exactly the Stasheff compactification when n = 1 or m = 1. The boundary strata of this compactification are products of the spaces K(m, n) and of the Stasheff polyhedra. Then we construct a dg Lie algebra naturally associat...

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves Mg,n. Their compactification Mg,n is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmüller space by the action of the mapping class group gives a compactification of Mg,n. We put an analytic structure on this quotient and prove that ...

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank 1 spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or qua...

In a classic paper, Smirnov [14] characterized the poset of compactifications of a completely regular space in terms of the proximities on the space. Later, Smyth [15] introduced the notion of a stable compactification of a T0-space and described them in terms of quasi-proximities on the space. Banaschewski [1] formulated Smirnov’s results in the pointfree setting, defining a compactification o...