435 research outputs found
Tropicalization of classical moduli spaces
The image of the complement of a hyperplane arrangement under a monomial map
can be tropicalized combinatorially using matroid theory. We apply this to
classical moduli spaces that are associated with complex reflection
arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa
quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our
primary example is the Burkhardt quartic, whose tropicalization is a
3-dimensional fan in 39-dimensional space. This effectuates a synthesis of
concrete and abstract approaches to tropical moduli of genus 2 curves.Comment: 33 page
Moduli spaces of rational weighted stable curves and tropical geometry
We study moduli spaces of rational weighted stable tropical curves, and their
connections with the classical Hassett spaces. Given a vector w of weights, the
moduli space of tropical w-stable curves can be given the structure of a
balanced fan if and only if w has only heavy and light entries. In this case,
we can express the moduli space as the Bergman fan of a graphic matroid.
Furthermore, we realize the tropical moduli space as a geometric
tropicalization, and as a Berkovich skeleton, of the classical moduli space.
This builds on previous work of Tevelev, Gibney--Maclagan, and
Abramovich--Caporaso--Payne. Finally, we construct the moduli spaces of
heavy/light weighted tropical curves as fiber products of unweighted spaces,
and explore parallels with the algebraic world.Comment: 26 pages, 8 TikZ figures. v3: Minor changes and corrections. Final
version to appear in Forum of Mathematics, Sigm
Tropicalization of the moduli space of stable maps
Let be an algebraic variety and let be a tropical variety associated
to . We study the tropicalization map from the moduli space of stable maps
into to the moduli space of tropical curves in . We prove that it is a
continuous map and that its image is compact and polyhedral. Loosely speaking,
when we deform algebraic curves in , the associated tropical curves in
deform continuously; moreover, the locus of realizable tropical curves inside
the space of all tropical curves is compact and polyhedral. Our main tools are
Berkovich spaces, formal models, balancing conditions, vanishing cycles and
quantifier elimination for rigid subanalytic sets.Comment: I improved the theorems using parametrized tropical curves in
Mathematische Zeitschrift, 201
The tropicalization of the moduli space of curves
We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of
stable curves is naturally identified with the moduli space of extended
tropical curves, and that this is compatible with the "naive" set-theoretic
tropicalization map. The proof passes through general structure results on the
skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct
tautological forgetful, clutching, and gluing maps between moduli spaces of
extended tropical curves and show that they are compatible with the analogous
tautological maps in the algebraic setting.Comment: v2: 55 pages. Expanded Section 2 with improved treatment of the
category of generalized cone complexes. Clarified the role of the coarse
moduli space and its analytification in the construction of the skeleton for
a toroidal DM stac
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
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
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