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...e 0 −→ T 1 Fα1 α2 (ξ )(F) −→ T 1 F X c α,g,n(ξ)(F) −→ E xt 1 T/S ˆ (ΩT ˆ/S ,π ∗ SF)−→ b −→ T 2 F α1 α2 (ξ )(F) −→ T 2 F X α,g,n(ξ)(F) −→ 0. Note that Ext1 T/S ˆ (ΩT/S ˆ ,π∗ SF) ∼ = F. It follows from =-=[DM]-=- that Ext1 T/S ˆ (ΩT/S ˆ ,Oˆ T ) is isomorphic to the pull back of the Q-invertible sheaf OMX (Lα1α2) andthe α,g,n homomorphism b is induced by the differential of the defining equation fα1α2. Now let...

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...assume all algebraic stacks (in particular all algebraic spaces and all schemes) to be quasi-separated and locally of finite type over k. A Deligne-Mumford stack is an algebraic stack in the sense of =-=[5]-=-, in other words an algebraic stack with unramified diagonal. For a Deligne-Mumford stack X we denote by X fl the big fppf-site and by X 'et the small 'etale site of X . The associated topoi of sheave...

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...W2 = .W2 x xt 2 C' x and although these families are fibrewise isomorphic, the isomorphism is not unique and may not globalize. To deal with this, one must use some variant of the ideal of stack (cf. =-=[D-M]-=-, §4). The most natural thing is to replace the normalization of X„ X x Xp by a scheme Xa p which must be given as part of the data and cannot be derived from the rest. _Yap should map to Xa Xx X0 and...

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... in putting this paper together. 2. Generalities on stacks 2.1. Criteria for a Deligne–Mumford stack. We refer the reader to [Ar] and [L-MB] for a general discussion of algebraic stacks (generalizing =-=[D-M]-=-), and to the appendix in [Vi] for anSTABLE MAPS 3 introduction. We spell out the conditions here, as we will follow them closely in the paper. We are given a category X along with a functor X → Sch/...

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...XG = (X × U)/G are schemes. Here G acts freely on U (which is an open subset of a representation of G) and hence on X × U. In the category of algebraic spaces, quotients of free actions always exist (=-=[D-M]-=- or [Ar]; see Proposition 22). By working with algebraic spaces we can therefore avoid the hypotheses of Proposition 23. Another reason to work with algebraic spaces comes from a theorem of Kollár and...

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...osition of the Kac-Moody groups theory [Ku, Sl]. 3. The stack SL r (O)nSL r (K)=SL r (AX ) Stacks We will need a few properties of stacks. Rather than giving formal definitions (for which we refer to =-=[D-M]-=- and especially [L-MB]), we will try here to give a rough idea of what stacks are and what they are good for. Many geometric objects (like vector bundles on a fixed variety, or varieties of a given ty...