In this work we present an exhaustive description, up to projective isomorphism, of all irreducible sextic curves in P 2 having a singular point of type A n , n ≥ 15, only rational singularities and global Milnor number at least 18. Moreover, we have developed a method for an explicit construction of sextic curves with at least eight-possibly infinitely near-double points. This method allows one to express such sextic curves in terms of arrangements of curves with lower degrees and it provides a geometric picture of possible deformations. Because of the large number of cases, we have chosen to carry out only a few to give some insights into the general situation.
We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in P 2 . Such a pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over Q( √ 5).
Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements are presented and developed for combinatorics with only double and triple points. This is part of a more general project to better understand the relationship between topology and combinatorics of line arrangements.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.