Tropical linear series, combinatorial flag arrangements and applications to the study of Weierstrass points

Overview & Synopsis

We first introduce new combinatorial objects called “matricubes”, a natural generalization of matroids. In the same way that matroids provide a combinatorial axiomatization of hyperplane arrangements in a vector space, matricubes abstract arrangements of flags. As for matroids, we provide cryptomorphic definitions of matricubes in terms of rank function, flats, circuits, and independent sets. We provide precise connections between matricubes and permutation arrays, and propose a description of matricubes in terms of local matroids.We then use matricubes to develop a purely combinatorial theory of limit linear series on metric graphs. This is based as well on the formalism of slope structures, which constrains the slopes of tropical meromorphic functions. We show that combinatorial linear series arise naturally by tropicalizing linear series on algebraic curves. We explore their topological properties and develop tools to study them. We provide a full classification of combinatorial linear series of rank one, showing that they are in one-to-one correspondence with harmonic morphisms from the graph to metric trees. This entails a smoothing theorem.Finally, we study tropical Weierstrass points, which are analogues, on tropical curves, of ramification points of line bundles on algebraic curves. The tropical Weierstrass locus of a divisor on a metric graph can be infinite. Nevertheless, we associate intrinsic weights to each of its connected components. We prove that the total weight of the tropical Weierstrass locus depends only on the degree of the divisor, its rank, and the genus of the tropical curve. Furthermore, in the case the metric graph is the tropicalization of an algebraic curve, we show, using combinatorial linear series, that our weights count the number of algebraic Weierstrass points which are tropicalized to each connected component of the tropical locus.In each of these contributions, we discuss multiple examples and ask open questions leading to other perspectives of research.