661 research outputs found
Lifting matroid divisors on tropical curves
Tropical geometry gives a bound on the ranks of divisors on curves in terms
of the combinatorics of the dual graph of a degeneration. We show that for a
family of examples, curves realizing this bound might only exist over certain
characteristics or over certain fields of definition. Our examples also apply
to the theory of metrized complexes and weighted graphs. These examples arise
by relating the lifting problem to matroid realizability. We also give a proof
of Mn\"ev universality with explicit bounds on the size of the matroid, which
may be of independent interest.Comment: 27 pages, 7 figures, final submitted version: several proofs
clarified and various minor change
A bit of tropical geometry
This friendly introduction to tropical geometry is meant to be accessible to
first year students in mathematics. The topics discussed here are basic
tropical algebra, tropical plane curves, some tropical intersections, and
Viro's patchworking. Each definition is explained with concrete examples and
illustrations. To a great exten, this text is an updated of a translation from
a french text by the first author. There is also a newly added section
highlighting new developments and perspectives on tropical geometry. In
addition, the final section provides an extensive list of references on the
subject.Comment: 27 pages, 19 figure
Lifting harmonic morphisms II: tropical curves and metrized complexes
In this paper we prove several lifting theorems for morphisms of tropical
curves. We interpret the obstruction to lifting a finite harmonic morphism of
augmented metric graphs to a morphism of algebraic curves as the non-vanishing
of certain Hurwitz numbers, and we give various conditions under which this
obstruction does vanish. In particular we show that any finite harmonic
morphism of (non-augmented) metric graphs lifts. We also give various
applications of these results. For example, we show that linear equivalence of
divisors on a tropical curve C coincides with the equivalence relation
generated by declaring that the fibers of every finite harmonic morphism from C
to the tropical projective line are equivalent. We study liftability of
metrized complexes equipped with a finite group action, and use this to
classify all augmented metric graphs arising as the tropicalization of a
hyperelliptic curve. We prove that there exists a d-gonal tropical curve that
does not lift to a d-gonal algebraic curve.
This article is the second in a series of two.Comment: 35 pages, 18 figures. This article used to be the second half of
arXiv:1303.4812, and is now its seque
On realizability of lines on tropical cubic surfaces and the Brundu-Logar normal form
We present results on the relative realizability of infinite families of
lines on general smooth tropical cubic surfaces. Inspired by the problem of
relative realizability of lines on surfaces, we investigate the information we
can derive tropically from the Brundu-Logar normal form of smooth cubic
surfaces. In particular, we prove that for a residue field of characteristic
the tropicalization of the Brundu-Logar normal form is not smooth. We
also take first steps in investigating the behavior of the tropicalized lines.Comment: 21 pages, 8 figure
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