Combinatorial Patchworking, Real Tropical Curves and Hyperbolic Varieties

Real algebraic geometry is the study of geometric objects defined by polynomial equations with real coefficients. This domain has connections with many areas of mathematics, such as analytic geometry, algebraic topology and analysis, as well as many applications in interdisciplinary fields such as computer-aided design, optimisation, computer vision and robotics. One particularly interesting class of geometric objects for these fields are the hyperbolic varieties, which admits a set of real points/lines/etc where all real lines/planes/etc through this set intersect the variety in a maximal number of real points. In this thesis, the so-called hyperbolic del Pezzo surfaces are classified by checking the hyperbolicity of some particular curves lying on those surfaces. Several properties of families of real curves are studied from the “logarithmic limit”, which is a particular unbounded graph with straight edges equipped with a “real structure”. We call those limits “real tropical curves”. Using this method, a combinatorial characterisation of the hyperbolicity loci of families of real curves is obtained, as well as several combinatorial criteria for a real curve to have some prescribed number of real components. Finally, some new counter-examples to a conjecture on the arrangement of those real components inside the plane are constructed