Counting Weierstrass points on degenerating algebraic curves

Weierstrass points on algebraic curves are special points of high importance in algebraic geometry and arithmetic geometry. In this talk, we study how those special points behave when the algebraic curve degenerates to a nodal curve. To this end, we first explain why tropical geometry is a relevant formalism for studying degeneration questions. We then define a tropical analogue on metric graphs (seen as a tropical curve) for Weierstrass points, and explore the properties of the so-called “tropical Weierstrass locus”. We also associate intrinsic weights to the connected components of this locus, and show that their total sum for a given tropical curve and divisor is a function of few combinatorial parameters (degree and rank of the divisor, genus of the metric graph). Finally, in the case the divisor on the metric graph comes from the tropicalization of a divisor on an algebraic curve, we specify the compatibility between the Weierstrass loci.