Modular compactifications of M_{1,n}

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne-Mumford stability. For every pair of integers 0<m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne-Mumford stack. We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks. In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for M_{1,n}.