ID: 1105.0509

Implicitization of surfaces via geometric tropicalization

May 3, 2011

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Geometries enumeratives complexe, reelle et tropicale

July 31, 2008

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Erwan Brugallé
Algebraic Geometry
Combinatorics

This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be understandable by a first year master student.

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Enumeration of Complex and Real Surfaces via Tropical Geometry

March 30, 2015

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Hannah Markwig, Thomas Markwig, Eugenii Shustin
Algebraic Geometry

We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin's lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in real three space. As application we show that there are pencils of rea...

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Tropical compactification in log-regular varieties

September 16, 2013

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Martin Ulirsch
Algebraic Geometry

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.

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Nonarchimedean geometry, tropicalization, and metrics on curves

April 2, 2011

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Matthew Baker, Sam Payne, Joseph Rabinoff
Algebraic Geometry
Number Theory

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the ana...

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Kontsevich's formula and the WDVV equations in tropical geometry

September 27, 2005

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Andreas Gathmann, Hannah Markwig
Algebraic Geometry

Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d-1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invaria...

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Projections of tropical varieties and their self-intersections

November 27, 2009

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Kerstin Hept, Thorsten Theobald
Algebraic Geometry
Combinatorics

We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally le...

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Polyhedral structures on tropical varieties

February 21, 2013

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Diane Maclagan
Commutative Algebra

Tropical varieties are polyhedral shadows of classical varieties. The purpose of these expository notes is to explain the origin of this polyhedral complex structure from the perspective of Gr\"obner bases. To appear in the proceedings of the 2011 Bellairs Workshop in Number Theory.

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The numbers of tropical plane curves through points in general position

April 19, 2005

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Andreas Gathmann, Hannah Markwig
Algebraic Geometry

We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but the only known proof was indirect with the help of Mikhalkin's Correspondence Theorem that translates this question into the well-known fact that the numbers of complex curves in a to...

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Tropical Invariants from the Secondary Fan

April 12, 2006

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Eric Katz
Algebraic Geometry
Combinatorics

In this paper, we consider weighted counts of tropical plane curves of particular combinatorial type through a certain number of generic points. We give a criterion, derived from tropical intersection theory on the secondary fan, for a weighted count to give a number invariant of the position of the points. By computing a certain intersection multiplicity, we show how Mikhalkin's approach to computing Gromov-Witten invariants fits into our approach. This begins to address a q...

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A guide to tropicalizations

August 31, 2011

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Walter Gubler
Algebraic Geometry

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis.

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