On the automorphisms of Hassett’s moduli spaces

Let Mg,A[n] be the moduli stack parametrizing weighted stable curves, and let Mg,A[n] be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of Mg,n and Mg,n, respectively, by assigning rational weights A = (a1, …, an), 0 < ai ≤ 1 to the markings. In particular, the classical Deligne-Mumford compactification arises for a1 = · · · = an = 1. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of M0,n developed by M. Kapranov, while in higher genus they may be related to the LMMP on Mg,n. We compute the automorphism groups of most of the Hassett spaces appearing in Kapranov’s blow-up construction. Furthermore, if g ≥ 1 we compute the automorphism groups of all Hassett spaces. In particular, we prove that if g ≥ 1 and 2g − 2 +n ≥ 3, then the automorphism groups of both Mg,A[n] and Mg,A[n] are isomorphic to a subgroup of Sn whose elements are permutations preserving the weight data in a suitable sense.