Positivity properties of toric line bundles and tropical divisors

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An algebraic variety is a subset of space that is given as the set of points where finitely many polynomial equations vanish. Intrinsic in this definition is an interplay between geometric properties of the variety and algebraic properties of the defining equations, a classical and fundamental problem in algebraic geometry. The projects described in this proposal will provide a major step towards the understanding of this relationship. Hilbert realised that a powerful way of studying the defining equations of a variety is by considering the so-called higher syzygies. A first syzygy is a polynomial relations satisfied by the given equations. There are finitely many first syzygies that generate (allowing polynomial coefficients) all first syzygies of a given set of equations. Now one can continue by looking at the relations between the first syzygies, the so-called second syzygies, and so on. The Betti numbers record the number of ith syzygies of a certain degree needed to generate all syzygies. They carry a lot of information about the embedding of the variety. For example, given the equations vanishing on seven points in space, one can detect from their Betti numbers whether there exists a polynomial of degree 3 vanishing on all of them. Usually one studies algebraic varieties from the abstract point of view, admitting many different embeddings into space that correspond to so-called very ample divisors, certain formal sums of subvarieties of one less dimension. It is a fundamental question in algebraic geometry to study the relationship between geometric properties of these divisors and algebraic properties of the resulting embedding. An example of a geometric property would be in how many points the divisor intersects an arbitrary given curve, and an example of an algebraic property would be that all defining equations can be generated from equations of degree two. This proposal deals with investigating this relationship using methods of discrete geometry.