On Sextic Curves with Big Milnor Number

Abstract

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.

Zariski $k$-plets via dessins d'enfants

“…Note that Condition (2) in the definition differs from paper to paper, the most common being the requirement that the pairs .P 2 ; C i / (or complements P 2 X C i ) should not be homeomorphic. In this paper, we choose equisingular deformation equivalence, i.e., being in the same component of the moduli space, as it is the strongest topologically meaningful 'global' equivalence relation.…”

“…Indeed, may have at most two orientations satisfying 4.1 (2), and if has at least one essential vertex, its orientation is uniquely determined by 4.1 (4)-.6/. Note also that each connected component of an admissible graph does have essential vertices (of all three kinds), as otherwise any component of @S 2 would be an oriented monochrome cycle.…”

“…Let D .B/. After a small deformation, we can assume that satisfies the general position assumptions 4.4.4 (1) and (2). Then, if B is not maximal, has a region R whose boundary contains at least two -vertices, and these two vertices can be brought together within R. This degeneration of results in a nontrivial degeneration of the curve.…”

Singular Points of Real Sextic Curves I

“…By referring to [2], we know that the family (y − x 2 ) 2 + ax(y − x 2 ) 2 + (y − x 2 ) 2 (by + cx 2 ) + xy(y − x 2 )(dy + ex 2 ) + y 2 (y − x 2 )(fy + gx 2 ) + hxy 3 (y − x 2 ) + y 4 (jy + kx 2 ) + lxy 5 + my 6 = 0 contains a curve with an A 19 singular point. However, it is interesting to note that this singular point cannot be obtained by our usual Maple computations [9].…”

“…a = 1, 2, 3,4,5,6,7,8,9,10 (a, b) = (2, 5 2 ), (2,3), (2, 7 2 ), (2,4), (2, 9 2 ), (2,5), (3, 7 2 ), (3,4) Y …”