- Tropical Intersection Products and Families of Tropical Curves (2012)
- This thesis is devoted to furthering the tropical intersection theory as well as to applying the developed theory to gain new insights about tropical moduli spaces. We use piecewise polynomials to define tropical cocycles that generalise the notion of tropical Cartier divisors to higher codimensions, introduce an intersection product of cocycles with tropical cycles and use the connection to toric geometry to prove a Poincaré duality for certain cases. Our main application of this Poincaré duality is the construction of intersection-theoretic fibres under a large class of tropical morphisms. We construct an intersection product of cycles on matroid varieties which are a natural generalisation of tropicalisations of classical linear spaces and the local blocks of smooth tropical varieties. The key ingredient is the ability to express a matroid variety contained in another matroid variety by a piecewise polynomial that is given in terms of the rank functions of the corresponding matroids. In particular, this enables us to intersect cycles on the moduli spaces of n-marked abstract rational curves. We also construct a pull-back of cycles along morphisms of smooth varieties, relate pull-backs to tropical modifications and show that every cycle on a matroid variety is rationally equivalent to its recession cycle and can be cut out by a cocycle. Finally, we define families of smooth rational tropical curves over smooth varieties and construct a tropical fibre product in order to show that every morphism of a smooth variety to the moduli space of abstract rational tropical curves induces a family of curves over the domain of the morphism. This leads to an alternative, inductive way of constructing moduli spaces of rational curves.