279 research outputs found
Tropical compactification in log-regular varieties
In this article we define a natural tropicalization procedure for closed
subsets of log-regular varieties in the case of constant coefficients and study
its basic properties. This framework allows us to generalize some of Tevelev's
results on tropical compactification as well as Hacking's result on the
cohomology of the link of a tropical variety to log-regular varieties.Comment: 15 pages; fixed an index mistake in Thm. 1.4 and Cor. 1.5 and
improved the exposition; to appear in Math. Zeitschrif
Functorial tropicalization of logarithmic schemes: The case of constant coefficients
The purpose of this article is to develop foundational techniques from
logarithmic geometry in order to define a functorial tropicalization map for
fine and saturated logarithmic schemes in the case of constant coefficients.
Our approach crucially uses the theory of fans in the sense of K. Kato and
generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the
toroidal case. For the convenience of the reader many examples as well as an
introductory treatment of the theory of Kato fans are included.Comment: v4: 33 pages. Restructured introduction, otherwise minor changes. To
appear in the Proceedings of the LM
Tropical geometry of moduli spaces of weighted stable curves
Hassett's moduli spaces of weighted stable curves form an important class of
alternate modular compactifications of the moduli space of smooth curves with
marked points. In this article we define a tropical analogue of these moduli
spaces and show that the naive set-theoretic tropicalization map can be
identified with a natural deformation retraction onto the non-Archimedean
skeleton. This result generalizes work of Abramovich, Caporaso, and Payne
treating the Deligne-Knudsen-Mumford compactification of the moduli space of
smooth curves with marked points. We also study tropical analogues of the
tautological maps, investigate the dependence of the tropical moduli spaces on
the weight data, and consider the example of Losev-Manin spaces.Comment: 25 pages, minor revisions, added further pictures as well as Section
6.2 discussing a tropical analogue of Kapranov's construction of
\Mbar_{0,n}. To appear in the Journal of the London Mathematical Societ
Logarithmic Picard groups, chip firing, and the combinatorial rank
Illusie has suggested that one should think of the classifying group of M_X^{gp}-torsors on a logarithmically smooth curve over a standard logarithmic point as a logarithmic analogue of the Picard group of . This logarithmic Picard group arises naturally as a quotient of the algebraic Picard group by lifts of the chip firing relations of the associated dual graph. We connect this perspective to Baker and Norine’s theory of ranks of divisors on a finite graph, and to Amini and Baker’s metrized complexes of curves. Moreover, we propose a definition of a combinatorial rank for line bundles on and prove that an analogue of the Riemann–Roch formula holds for our combinatorial rank. Our proof proceeds by carefully describing the relationship between the logarithmic Picard group on a logarithmic curve and the Picard group of the associated metrized complex. This approach suggests a natural categorical framework for metrized complexes, namely the category of logarithmic curves
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