an:06355716
Zbl 1321.14025
Abramovich, Dan; Chen, Qile
Stable logarithmic maps to Deligne-Faltings pairs. II
EN
Asian J. Math. 18, No. 3, 465-488 (2014).
00337018
2014
j
14H10 14N35 14D23 14A20
moduli spaces; logarithmic structures
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 [\textit{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 [\textit{M. Gross} and \textit{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).
Dawei Chen (Chestnut Hill)
Zbl 1311.14028; Zbl 1281.14044