The automorphism group of $\overline{M}_{0,n}$

Abstract

IntroductionThe moduli space M g,n of smooth n-pointed curves of genus g, and its projective closure, the Deligne-Mumford compactification M g,n , is a classical object of study that reflects many of the properties of families of pointed curves. As a matter of fact, the study of its biregular geometry is of interest in itself and has become a central theme in various areas of mathematics.Already for small n, the moduli spaces M 0,n are quite intricate objects deeply rooted in classical algebraic geometry. Under this perspective, Kapranov showed in [Ka] that M 0,n is identified with the closure of the subscheme of the Hilbert scheme parametrizing rational normal curves passing through n points in linearly general position in P n−2 . Via this identification, given n − 1 points in linearly general position in P n−3 , M 0,n is isomorphic to an iterated blow-up of P n−3 at the strict transforms of all the linear spaces spanned by subsets of the points in order of increasing dimension. In a natural way, then, base point free linear systems on M 0,n are identified with linear systems on P n−3 whose base locus is quite special and supported on so-called vital spaces, i.e. spans of subsets of the given points. Another feature of this picture is that all these vital spaces correspond to divisors in M 0,n which have a modular interpretation as products of M 0,r for r < n. In this interpretation, the modular forgetful maps φ I : M 0,n → M 0,n−|I| , which forget points indexed by I ⊂ {1, . . . , n}, correspond, up to standard Cremona transformations, to linear projections from vital spaces. The aim of this paper is to study automorphisms of M 0,n with the aid of Kapranov's beautiful description.It is expected that the only possible biregular automorphisms of M 0,n are the one associated to a permutation of the markings. Any such morphism has to permute the forgetful maps onto M 0,n−1 as well. This induces, on P n−3 , special birational maps that switch lines through n − 1 points in general position. On the other hand if we were able to prove that any automorphism has to permute forgetful maps this should lead to a proof that every automorphism is a permutation. Our main tool to classify Aut(M 0,n ) is therefore the following Theorem.Theorem 1. Let f : M 0,n → M 0,r1 × . . . × M 0,r h be a dominant morphism with connected fibers. Then f is a forgetful map.

The automorphism group of M̄g,n

“…Such a clear description of the fibrations ofM g,n is no longer true for g = 1. An explicit counterexample to this fact was given by R. Pandharipande and can be found in [5,Example A.2]. See also [18] for similar constructions.…”

“…Finally, let us consider the case g = 0. It is well known that any fibrationM 0,5 → M 0,4 factorizes through a forgetful morphism, see for instance [5]. This yields a surjective homomorphism of groups χ : Aut(M 0,5 ) −→ S 5 .…”

“…Recently, in [4,5], Bruno and Mella studied the fibrations ofM 0,n using its description as the closure of the subscheme of the Hilbert scheme parametrizing rational normal curves passing through n points in linearly general position in P n−2 given by M. Kapranov in [13]. It was expected that the only possible biregular automorphisms ofM 0,n were the ones associated to a permutation of the markings.…”

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LetMg,n be the moduli stack parametrizing Deligne-Mumford stable n-pointed genus g curves and letM g,n be its coarse moduli space: the Deligne-Mumford compactification of the moduli space of n-pointed genus g smooth curves. We prove that the automorphism groups ofM g,n andMg,n are isomorphic to the symmetric group on n elements Sn for any g, n such that 2g − 2 + n 3, and compute the remaining cases.on fibrations derived that the automorphism group ofM 0,n is the symmetric group S n for any n 5 [5, Theorem 4.3].The aim of this work is to extend [5, Theorem 4.3] to arbitrary values of g, n and to the stackM g,n . In extending [5, Theorem 4.3], we will indeed reprove Bruno and Mella's result with a different method.

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“…on fibrations derived that the automorphism group ofM 0,n is the symmetric group S n for any n 5 [5, Theorem 4.3].…”