How to tell the Maroni invariants and the scrollar ruling degrees from the Newton polygon

Géométrie des espaces singuliers

Salle Duhem M3
Univ. Lille 1
Mardi, 23 Mai, 2017 - 10:15 - 11:15
Let $C$ be a non-hyperelliptic curve of genus $g > 2$ along with a degree $d$ morphism $\phi : C \to P^1$. If C is canonically embedded, then the fibers of phi naturally sweep out a rational normal scroll $S$ which comes equipped with invariants $e_1, ..., e_{d-1}$ called the Maroni invariants of $C$. Inside the scroll our curve is cut out by $(d^2-3d)/2$ quadrics, each of which comes along with a 'ruling degree', denoted by $ b_1, ..., b_{(d^2-3d)/2}$. The invariants $e_i$ and $b_i$ play an important role in the study of linear systems on algebraic curves.
In this talk we will consider the case where $C$ is defined by a sufficiently generic polynomial $f(x,y) = 0$ having a prescribed Newton polygon $\Delta$, and where $\phi : C \to P^1: (x,y) \to x^ay^b$ is a monomial map with $\gcd(a,b) = 1$. We will show how to tell the Maroni invariants $e_i$ by merely looking at $\Delta$, and under some combinatorial constraints on $\Delta$ we will give a similar interpretation for the scrollar ruling degrees $b_i$. Our main focus will be on the cases $d = 3$ and $d = 4$. In the latter case we will show that the scrollar ruling degrees $b_1$ and $b_2$ actually arise as the Maroni invariants of the cubic resolvent (= Recillas' trigonal construction).
Most of this talk will present joint work with Filip Cools. At the end it will cover joint work in progress with Yongqiang Zhao.