Algebraic Spaces

Abstract

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Nagata compactification for algebraic spaces

“…Aside from using a couple of self-contained results (Lemma 3.1.4 and Proposition A.1.3), our proof uses Corollary 3.1.12, which rests on Theorem 1.2.2, whose proof in turn relies on almost everything in §3.1 and §A.3. However, the noetherian case of Corollary 3.1.12 is an old result of Knutson [K,III,Thm. 3.3].…”

“…Let S be any algebraic space. In [K,II,§6] the concept of points of S and the associated topological space |S| is defined in general but is only developed in a substantial manner when S is quasi-separated (i.e., ∆ S/ Spec Z is quasi-compact). The key to the theory is the fact [K,II,6.2] that if S is quasi-separated and s : Spec k → S is a morphism with k a field then there is a unique subfield k 0 ⊆ k such that s factors through a (necessarily unique) monomorphism s 0 : Spec k 0 → S. This can fail when S is not quasi-separated, as the following example shows, so the theory of points of S needs to be modified in general.…”

Global structures for the moduli of (punctured) super riemann surfaces

“…The representability of the moduli functor of ordinary curves of genus g with a level-n structure means that there is a moduli scheme M and a universal curve XM -+ M. Remark 3.6. For every trivializing covering p : Us ---> Ms, the fibred product R = Us x Ms Us defines an &ale equivalence relation (Pl, P2) : Us Xm, Us=~Us, whose quotient (in the category of locally ringed spaces [ 16]) is Ms. Then, the functor of spin curves 8spin ~ Ms e is the quotient of equivalence relation R" _~ Us" x m," Us" :::¢ Us" in the category of sheaves of sets.…”

“…But even in the category of schemes, 6tale equivalence relations may fail to have a categorical quotient [16]. This problem is solved with the introduction of Artin's algebraic spaces [1,16], which are natural outgrows of schemes. In this larger category any &ale equivalence relation has a categorical quotient [ 16].…”

“…Algebraic superspaces are defined here for the first time, and they are the graded objects corresponding to Artin's algebraic spaces [1,16]. As Deligne suggested, the category of superschemes, that is, schemes with a graded structure in the sense of Berezin-Leites-Kostant, is not wide enough to contain the supermoduli spaces of SUSY curves.…”