Related papers: Counting Algebraic Curves with Tropical Geometry
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…
We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…
In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
We construct algebraic curves in abelian surfaces starting from tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface.…
We discuss, following Mikhalkin, Brugall\'e, and many others, the counting of curves on toric surfaces with prescribed genus, Newton polygon, and intersection pattern with the toric boundary divisor, both at assigned and unassigned points.…
We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed $j$-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts…
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the…
We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4-8, 2011, and at the ELGA 2011 school on Algebraic Geometry and…
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…
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
Counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$ with fixed contact order with the toric boundary and satisfying point conditions can be determined with tropical methods by Mikhalkin. If we require that our curves intersect the zero-…
Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization…