Title: Repairing tropical curves by means of tropical modifications



Abstract:

Tropical geometry is a piecewise-linear shadow of algebraic geometry that preserves important geometric invariants. Often, we can derive classical statements from these (easier) combinatorial objects. One general difficulty in this approach is that tropicalization strongly depends on the embedding of the algebraic variety. Thus, the task of finding a suitable embedding or of repairing a given "bad" embedding to obtain a nicer tropicalization that better reflects the geometry of the input object becomes essential for many applications. In this talk, I will show how to use linear tropical modifications and Berkovich skeleta to achieve such goal in the curve case. Our motivating examples will be plane elliptic cubics and genus two hyperelliptic curves. Based on joint work with Hannah Markwig (arXiv:1409.7430) and ongoing work in progress with Hannah Markwig and Ralph Morrison.