Diego Antonio Robayo Bargans

• This project originates in the divisor theory of discrete graphs and (abstract) tropical curves, and more precisely in their geometric gonality. The notion of geometric gonality concerns the degree of harmonic morphisms from tropical modifications of the source graph to a tree. We describe a tropical intersection theoretic framework that includes moduli spaces of tropical curves of positive genus (and possibly marked) motivated by a recent (at the time of the project) combinatorial proof of the following result: Every genus-\( g \) graph has a tropical modification that is the source of a degree-\( (\lceil\frac{g}{2}\rceil+1) \) tropical cover of a tree. The design of our framework not only furnishes a different proof of the previous result but also provides enough leeway for a systematic argument that generalizes the previous methods. In particular, after fixing the degree \( d \) of the cover, the marking \( m \) and genus \( h \) of the target, and the ramification profiles above the marked ends we are able to study the loci of marked tropical curves that are the source of a tropical cover with these conditions. We arrive at these loci in two steps: first, we produce a tropical cycle in the corresponding moduli space of discrete admissible covers, then we pushforward this cycle through the corresponding source and forgetting-the-marking morphisms. Moreover, the first-mentioned result is just a consequence of the irreducibility of these moduli spaces (in a precise sense) and the special case of \( d=\lceil\frac{g}{2}\rceil+1 \), \( h=0 \), and \( m=3g \). Our framework does not only shed new light into the underlying structure of these loci, and their general behavior, but also provides immediate access to some enumerative computations concerning these cycles, giving new proofs and generalizations of known results.