Conference abstracts

Session S02 - Commutative Algebra and Algebraic Geometry

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Rational Harnack Curves on Toric Surfaces

Jorge Alberto Olarte Parra

Universidad de los Andes, Colombia   -

Harnack curves are a family of real algebraic curves who are distinguished because their topology is well understood, meaning that Hilbert's 16th problem is solved for these curves. Let $f$ be a 2 variable real polynomial whose Newton polygon is $\Delta$ and let $C$ be the curve defined as the zeros of $f$ inside the toric variety $X_\Delta$. The original definition of Harnack curves by Mikhalkin states that the real part of $C$, $\mathbb{R}C \subseteq \mathbb{R}X_\Delta$, is a Harnack curve if and only if the following conditions are satisfied:

1. The number of connected components of $\mathbb{R}C$ is maximal, that is $g+1$, where $g$ is the arithmethic genus of $C$.

2. Only one component $O$ intersects the axes of $\mathbb{R}X_\Delta$.

3. Let $l_1,\dots,l_n$ be the axes of $X_\Delta$ ordered in a way such that it agrees with the cyclical order of their corresponding sides of $\Delta$ and let $d_1,\dots,d_n$ be the integer lengths of the corresponding sides. Then $O$ can be divided into disjoint arcs $\alpha_1,\dots \alpha_n$ such that $\alpha_i \cap l_i = d_i$ and $\alpha_i\cap l_j = 0$ when $j\neq i$.

These curves have several different characterizations, for example, its amoeba (the image of $C$ under the map $(z,w) \mapsto (\log|z|, \log|w|)$) is of maximal area. These curves have applications to physics through dimer theory. In this poster we focus on rational Harnack curves, which are Harnack curves of genus 0 and we show how these curves can be explicitly parametrized using the homogeneous coordinates of $X_\Delta$.

Joint work with Mauricio Velasco (Universidad de los Andes, Colombia).

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