Diviseurs sur les courbes réelles

Abstract : In an article about sums of squares, SCHEIDERER proved that for every real, algebraic, projective, irreducible, smooth curve with some real points, their exists an integer N such that every divisor of degre not lower than N is linearly equivalent to a divisor whose support is totally real. Then HUISMAN and MONNIER proved that for real curves with many components, ie. those with at least as many components as the genus g, assumed here to be positive, of the curve, one can choose N equal to 2g − 1. MONNIER also dealed with singular curves: he showed that for some of them such an integer does not exist and gave some others where it does exist. In this thesis we extend the classe of singular curves for wich such an integer exists, essentially those with nodes and cusps, and we sometimes manage to bound such an integer in terms of the genus. To do so, an "iterative singularisation" is used and also a slightly different invariant where we ask the real points of the support to be distinct from each-other. We extend the results about curves with many components to that new invariant and deal with curves of genus 2 having only one component, which is the "very first" unknown case so far: in that case, 3 cannot bound the invariant, but 5 does.
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Submitted on : Monday, November 4, 2013 - 2:48:37 PM
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  • HAL Id : tel-00879645, version 1

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Alexandre Bardet. Diviseurs sur les courbes réelles. Mathématiques générales [math.GM]. Université d'Angers, 2013. Français. ⟨tel-00879645⟩

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