86 research outputs found
On fundamental groups of plane curve complements
In this paper we discuss some properties of fundamental groups and Alexander
polynomials of plane curves. We discuss the relationship of the non-triviality
of Alexander polynomials and the notion of (nearly) freeness for irreducible
plane curves. We reprove and restate in modern terms a somewhat forgotten
result of Zariski. Finally, we describe some topological properties of curves
with abelian fundamental group
Cartier and Weil Divisors on Varieties with Quotient Singularities
The main goal of this paper is to show that the notions of Weil and Cartier
-divisors coincide for -manifolds and give a procedure to
express a rational Weil divisor as a rational Cartier divisor. The theory is
illustrated on weighted projective spaces and weighted blow-ups.Comment: 16 page
On the connection between fundamental groups and pencils with multiple fibers
We present two results about the relationship between fundamental groups of
quasiprojective manifolds and linear systems on a projectivization. We prove
the existence of a plane curve with non-abelian fundamental group of the
complement which does not admit a mapping onto an orbifold with non-abelian
fundamental group. We also find an affine manifold whose irreducible components
of its characteristic varieties do not come from the pull-back of the
characteristic varieties of an orbifold
A topological invariant of line arrangements
We define a new topological invariant of line arrangements in the complex
projective plane. This invariant is a root of unity defined under some
combinatorial restrictions for arrangements endowed with some special torsion
character on the fundamental group of their complements. It is derived from the
peripheral structure on the group induced by the inclusion map of the boundary
of a tubular neigborhood in the exterior of the arrangement. By similarity with
knot theory, it can be viewed as an analogue of linking numbers. This is an
orientation-preserving invariant for ordered arrangements. We give an explicit
method to compute the invariant from the equations of the arrangement, by using
wiring diagrams introduced by Arvola, that encode the braid monodromy.
Moreover, this invariant is a crucial ingredient to compute the depth of a
character satisfying some resonant conditions, and complete the existent
methods by Libgober and the first author. Finally, we compute the invariant for
extended MacLane arrangements with an additional line and observe that it takes
different values for the deformation classes.Comment: 19 pages, 5 figure
Rational homology disk smoothings of surface singularities; the exceptional cases
It is known (Stipsicz-Szab\'o-Wahl) that there are exactly three
triply-infinite and seven singly-infinite families of weighted homogeneous
normal surface singularities admitting a rational homology disk
(HD) smoothing, i.e., having a Milnor fibre with Milnor number
zero. Some examples are found by an explicit "quotient construction", while
others require the "Pinkham method". The fundamental group of the Milnor fibre
has been known for all except the three exceptional families and . In this paper, we settle
these cases. We present a new explicit construction for the
family, showing the fundamental group is non-abelian (as occurred previously
only for the and cases).
We show that the fundamental groups for and are abelian, hence easily computed; using the Pinkham method here
requires precise calculations for the fundamental group of the complement of a
plane curve.Comment: 24 page
Arrangements of hypersurfaces and Bestvina-Brady groups
We show that quasi-projective Bestvina-Brady groups are fundamental groups of
complements to hyperplane arrangements. Furthermore we relate other normal
subgroups of right-angled Artin groups to complements to arrangements of
hypersurfaces. We thus obtain examples of hypersurface complements whose
fundamental groups satisfy various finiteness properties.Comment: 21 page
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