Group actions on stacks and applications

Matthieu Romagny
2005 The Michigan mathematical journal  
We provide the correct framework for the treatment of group actions on algebraic stacks (including fixed points and quotients). It is then used to exploit some natural actions on moduli stacks of maps of curves. This leads to the construction of a nice desingularization of the normal stack of curves with level structures considered by Deligne and Mumford ([DM]), and to a presentation of stacks of Galois covers of curves as quotients of a scheme by a finite group. The motivation at the origin of
more » ... this article is to investigate some ways in which one can construct moduli for curves and covers above them, using tools from stack theory. This idea arose from reading Bertin-Mézard [BM] (especially its §5) and Abramovich-Corti-Vistoli [ACV]. Our approach is in the spirit of most recent works where one uses the flexibility of the language of algebraic stacks. This language has two (twin) aspects, category-theoretic on one side, and geometric on the other side. Some of our arguments, especially in section 8, are formal arguments involving general constructions concerning group actions on algebraic stacks (this is more on the categoric side). They are, intrinsically, natural enough so they preserve the "modular" aspect. In trying to isolate these arguments, we were led to write results of independent interest. It seemed therefore more adequate to present them in a separated, self-contained part. Thus the article is split into two parts, of comparable size. More specifically, groups are ubiquitous in Algebraic Geometry (when one focuses on curves and maps between them, examples are : the automorphism group, fundamental group, monodromy group, permutation group of the ramification points...). It is natural to ask : Can we handle group actions on stacks in the same fashion as we do on schemes ? For example, we expect that the quotient of the stack of curves with ordered marked points M g,n by the symmetric group should classify curves with unordered marked points ; if G acts on a scheme X then the fixed points of the stack Pic(X) under G should be related to G-linearized line bundles on X ; the quotient of the modular stack-curve X 1 (N ) by (Z/N Z) × should be X 0 (N ) (the notations are hopefully well-known to the reader). Other important examples appear in the literature : action of tori on stacks of stable maps in Gromov-Witten theory [Ko] [GP], action of the symmetric group S d on a stack of multisections in [L-MB], (6.6). Our aim is to provide here the material necessary to handle the questions raised above and answer them, as well as to give other applications. Let us now explain in more detail the structure and the results of the article. In part A we discuss the notion of a group action on a stack. We are mainly interested in giving general conditions under which the fixed points and the quotient of an algebraic stack are algebraic. In sections 1 and 2 we give definitions and basics on actions. For simplicity let us now consider a flat group scheme G and an algebraic stack M, both of finite presentation (abbreviated fp) over some base scheme. Slightly sharper assumptions are in the text. In sections 3 and 4 we establish : • assume that the structure sheaf O G is locally free over the base. If moreover G is proper, or if M is a Deligne-Mumford stack, then there is an algebraic stack of fixed points M G . The map M G → M is representable and separated, with even better properties when G is finite (th. 3.3, prop. 3.7).
doi:10.1307/mmj/1114021093 fatcat:e4k54jx4hzex5jhdeq7gqaim5i