LOGARITHMIC GEOMETRY AND ALGEBRAIC STACKS

By Martin Olsson. 2003

188 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b9eb735707c5a692f1e279b638d543804e3b989

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Citations in the corpus

A MODULI STACK OF TROPICAL CURVES (2017, background): “ sense of AC(S)when S is not of ﬁnite type. Unfortunately, AC is not a stack in general. It does, at least, turn out to be a stack when C =σ is a cone, as the following lemma demonstrates. Lemma 6.3 ([Ols03] Proposition 5.17). Suppose that σ is a rational polyhedral cone. The stack Aσ is isomorphic ”
Logarithmic Structures of Fontaine-Illusie. II ---Logarithmic Flat Topology (2019, methodology): “e that T and S are fs log schemes, is satisﬁed. This is proved in the same way as the case of ﬁne log schemes. We prove the stability of log ﬂatness for compositions. This stability is also proved in [Ol] Corollary 4.12 (ii) by another method. See also Ogus’ book [Og] Chapter IV, Proposition 4.1.2 (”
Langlands correspondence for isocrystals and the existence of crystalline companions for curves (2013, background): “rtant. Otherwise, non-separated algebraic spaces appear in the simplicial space cosk0(X→X), and we are not able to apply Proposition 2.2.22. Non quasi-separated stacks naturally appear in the work of [O1]. (ii) We may treat the non quasi-compact case without much eﬀort. For a separated scheme Xlocal”
Logarithmic Geometry and Moduli (2010, methodology): “([46], 5.”
Using stacks to impose tangency conditions on curves (2003, background): “Moreover, there is an equivalence between pairs (L, s), with L an invertible sheaf on X and s ∈ Γ(X, L), and morphisms X → [A 1 /Gm], where Gm acts on A 1 by multiplication [ Ol1 , 5.13].”
(Log) twisted curves (2007, unknown): “From this and [Ols03a, 5.20] it follows that if we choose an ´etale morphism in some neighborhood of the image of p , where t ∈ O S , the stack C is isomorphic to Here t (cid:1) ∈ O S is a section such that t α r = t , and a scheme-valued point u ∈ µ α r acts by z (cid:5)→ uz and w (cid:5)→ u − 1 w…”
Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces (2007, background): “13 For basic properties of log structures we refer to [25] and [39].”
Stable logarithmic maps to Deligne-Faltings pairs I (2010, methodology): “X associated to Ddeﬁned in example A.1.2. Then MX forms a DF log structure on X, which corresponds to the line bundle OX(−D) and the natural inclusion OX(−D) →OX. A.3. Olsson’s Log Stacks. We follow [Ols03a] to introduce the algebraic stack parameterizing log structures. Let us ﬁx a base scheme S, a”
Skeletons and Fans of Logarithmic Structures (2015, methodology): “hamp) etale. The following lemma characterizes the etale site of Log as a category of presheaves: Lemma 5.4. (i)The Artin cones are an etale cover of Log. 1Our conventions dier from Olsson’s: In [Ols03], the stack Log parameterizes all ne logarithmic structures; to conform with our convention that a”
Decomposition of degenerate Gromov–Witten invariants (2017, unknown): “arated diagonal. An Artin stack logarithmically ´etale over Speckis called an Artin fan. The logarithmic structure of Xis encoded by a morphism X→ Log to Olsson’s stack Log of ﬁne log structures, see [Ol]. One crucial idea developed in the context of the present paper is a reﬁnement of the stack Log”
Étale dévissage, descent and pushouts of stacks (2010, background): “ and have separated diagonals. In the remainder of this appendix we give criteria for when this is the case. An important example of a stack with non-separated diagonal is the stack of log structures [Ols03]. On the other hand, this stack has at least locally separated diagonal. Deﬁnition (A.2). Let”