On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.

Position of singularities and fundamental group of the complement of a union of lines

“…Suppose t = 0; then Σ is r + 1 lines in general position and π 1 (CP 2 \ Σ) ∼ = A r , where A r is a free abelian group on r generators [7].…”

“…In [7], Zariski showed that the fundamental group of the complement of an irreducible sextic curve in the projective plane with six cusps lying on a conic is isomorphic to the free product of Z 2 and Z 3 (see also [4]). In [8], he then showed that the complement of an irreducible sextic curve with six cusps that do not lie on a conic has abelian fundamental group.…”

Plane curves and their fundamental groups: Generalizations of Uludağ’s construction

“…A different direction for the need of fundamental groups' computations is for getting more examples of Zariski pairs [36,37]. A pair of plane curves is called a Zariski pair if they have the same combinatorics (i.e.…”

Computable Isomorphism Invariants for the Fundamental Group of the Complement of a Plane Projective Curve

“…We let c, be an irreducible sextic with six cusps on a conic. In this case it can be shown (Zariski [6]) that ttx(P2 -cx) = <g" g2; g2, g|>. We let c2 be an irreducible nonsingular sextic.…”

“…In this case H = Z/4 Z and it can be shown that 77,(P2 -c,) = <g" g2; g\g22, g\, (gxg2?gx~2) (cf. Zariski [6]). We have the following result.…”