Fibonacci numbers and self-dual lattice structures for plane branches

Abstract

Consider a plane branch, that is, an irreducible germ of curve on a smooth complex analytic surface. We define its blow-up complexity as the number of blow-ups of points needed to achieve its minimal embedded resolution. We show that there are F 2n-4 topological types of blow-up complexity n, where F n is the n-th Fibonacci number. We introduce complexity-preserving operations on topological types which increase the multiplicity and we deduce that the maximal multiplicity for a plane branch of blow-up complexity n is F n. It is achieved by exactly two topological types, one of them being distinguished as the only type which maximizes the Milnor number. We show moreover that there exists a natural partial order relation on the set of topological types of plane branches of blow-up complexity n, making this set a distributive lattice. We prove that this lattice admits a unique order-inverting bijection. As this bijection is involutive, it defines a duality for topological types of plane branches. There are F n-2 self-dual topological types of blow-up complexity n. Our proofs are done by encoding the topological types by the associated Enriques diagrams.