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Stable logarithmic maps to Deligne-Faltings pairs. II. (English) Zbl 1321.14025
Let $$\mathcal{K}_{\Gamma}(Y)$$ be the stack parametrizing stable logarithmic maps of log-smooth curves into a logarithmic scheme $$Y$$ with the relevant numerical data $$\Gamma$$, such as genus, marked points, curve class and other indicators (contact orders), related to the logarithmic structure. It was proved in [Q. Chen, Ann. Math. (2) 180, No. 2, 455–521 (2014; Zbl 1311.14028)] that $$\mathcal{K}_{\Gamma}(Y)$$ is algebraic and proper when the logarithmic structure of $$Y$$ is given by a line bundle with a section, and more generally in [M. Gross and B. Siebert, J. Am. Math. Soc. 26, No. 2, 451–510 (2013; Zbl 1281.14044)]. The motivating case in [Zbl 1311.14028] is that of a pair $$(\underline{Y}, \underline{D})$$, where $$\underline{D}$$ is a smooth divisor in the smooth locus of the scheme $$\underline{Y}$$ underlying $$Y$$. Based on this special case, in the paper under review the authors observe that one can give a “pure-throught” proof of algebraicity and properness of the stack $$\mathcal{K}_{\Gamma}(Y)$$ whenever $$Y$$ is a Deligne-Faltings logarithmic structure (Theorem 2.6). This observation covers a number of the cases of interest, such as a variety with a simple normal crossings divisor, or a simple normal crossings degeneration of a variety with a simple normal crossings divisors. The authors further extend the result to some more general settings (Theorems 3.15 and 5.7).

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14D23 Stacks and moduli problems 14A20 Generalizations (algebraic spaces, stacks)
##### Keywords:
moduli spaces; logarithmic structures
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