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Tropical intersection theory, and real inflection points of real algebraic curves

Abstract : This thesis is divided in two main parts. First, we study the relationships between intersection theories in tropical and algebraic geometry. Then, we study the question of the possibilities for the distribution of the real inflection points associated to a real linear system defined on a smooth real algebraic curve. In the first part, we present new results linking algebraic and tropical intersection theories over a very-affine algebraic variety defined over a particular non-Archimedean field (known as Mal’cev-Newmann field). The main result concerns the intersection of a one-dimensional algebraic cycle with a Cartier divisor in a variety with simple tropicalization. In the second part, we obtain first a characterization of the distribution of real inflection points associated to a real complete linear system of degree d>1 defined over a smooth real elliptic curve. Then we study some canonical, non-hyperelliptic real algebraic curves of genus 4 in a 3-dimensional projective space. We obtain a formule that relies the amount of real Weierstrass points of such a curve with the Euler-Poincaré characteristic of certain topological space. Finally, using O. Viro’s Patch-working technique, we construct an example of a smooth, non-hyperelliptic real algebraic curve of genus 4 having 30 real Weierstrass points.
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Cristhian Emmanuel Garay-Lopez. Tropical intersection theory, and real inflection points of real algebraic curves. Algebraic Geometry [math.AG]. Université Pierre et Marie Curie - Paris VI, 2015. English. ⟨NNT : 2015PA066364⟩. ⟨tel-01267247⟩



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