Semistable Modular Compactifications of Moduli Spaces of Genus One Curves

We clarify the definition of an infinitesimal automorphism of a log smooth curve, and show that logarithmic structure is capable of fixing the underlying infinitesimal automorphisms on a semistable curve. We then introduce the notion of an m-stable partially aligned logarithmic curve and construct a few natural semistable modular compactifications of the moduli space of smooth genus one curves. Finally, we prove that our moduli space of m-stable partially aligned log curves resolves the indeterminacy between the moduli space of Deligne-Mumford stable curves and the moduli space of m-stable curves constructed by Smyth