English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Consider a plane branch, that is, an irreducible germ of curve on a smoothcomplex analytic surface. We define its blow-up complexity as the number ofblow-ups of points necessary to achieve its minimal embedded resolution. Weshow that there are $F_{2n-4}$ topological types of blow-up complexity $n$,where $F_{n}$ is the $n$-th Fibonacci number. We introducecomplexity-preserving operations on topological types which increase themultiplicity and we deduce that the maximal multiplicity for a plane branch ofblow-up complexity $n$ is $F_n$. It is achieved by exactly two topologicaltypes, one of them being distinguished as the only type which maximizes theMilnor number. We show moreover that there exists a natural partial orderrelation on the set of topological types of plane branches of blow-upcomplexity $n$, making this set a distributive lattice, that is, any two of itselements admit an infimum and a supremum, each one of these operations beeingdistributive relative to the second one. We prove that this lattice admits aunique order-inverting bijection. As this bijection is involutive, it defines aduality for topological types of plane branches. The type which maximizes theMilnor number is also the maximal element of this lattice and its dual is theunique type with minimal Milnor number. There are $F_{n-2}$ self-dualtopological types of blow-up complexity $n$. Our proofs are done by encodingthe topological types by the associated Enriques diagrams.

Fibonacci numbers and self-dual lattice structures for plane branches