Luca Battistella (Università di Bologna)

Abstract: Maps to projective space (up to projectivities) are given by basepoint-free linear series, thus these are key to understanding the extrinsic geometry of algebraic curves. In fact, the existence of special linear series is the cornerstone of curve theory, providing a plethora of cycles on the moduli space. But what happens to a linear series when the underlying curve degenerates and becomes nodal? A first answer was given by Eisenbud and Harris: their theory of limit linear series led to the proof of many foundational results in Brill--Noether theory, and the exploration of the birational geometry of the moduli space of curves. I will report on a joint work in progress with Francesca Carocci and Jonathan Wise, in which we review this question from a moduli-theoretic and logarithmic perspective. The combinatorics of the Bruhat--Tits building of PGL plays a prominent role informing the behaviour of vector bundles on tropical varieties (curves, in particular).