Chow hypersurfaces and realizability problems in tropical geometry.

Tropical geometry is an area of mathematics between algebraic geometry, polyhedral geometry and combinatorics. The basic principle of tropical geometry is to associate to an algebraic variety X a polyhedral complex Trop(X) called the tropicalization of X. The tropicalization of X can be studied by means of polyhedral geometry and combinatorics and reflects many properties of the original variety X. Given a projective variety X . Pn of codimension k+1, the Chow hypersurface ZX is the hypersurface of the Grassmannian Gr(k; n) parametrizing k-dimensional linear subspaces of Pn that intersect X. In Chapter 3 we introduce and describe a tropical Chow hypersurface Trop(ZX). This object only depends on the tropical variety Trop(X) and we provide an explicit way to obtain Trop(ZX) from Trop(X). Finally we prove that, when X is a curve in P3 , Trop(X) can be reconstructed from Trop(ZX). In Chapter 4 we study the geometric properties of Chow hypersurfaces of space curves and other special varieties in the Grassmannian Gr(1; P3 ), such as the Hurwitz hypersurface and the bitangent congruence of a space surface. We give new proofs for the bidegrees of the secant, bitangent and in inflectional congruences, using geometric techniques such as duality, polar loci and projections. We also study the singularities of these congruences. In Chapter 5 we start with a family of projective curves and we study the geometric properties of the fibers by looking at their tropicalization. Given a tropical curve Ʃ and a family of algebraic curves X we produce an algorithm to describe the locus of fibbers of X whose tropicalization contains Ʃ