162 research outputs found
On plane sextics with double singular points
We compute the fundamental groups of five maximizing sextics with double
singular points only; in four cases, the groups are as expected. The approach
used would apply to other sextics as well, given their equations.Comment: A few explanations and references adde
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 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
Number of Jordan blocks of the maximal size for local monodromies
We prove formulas for the number of Jordan blocks of the maximal size for
local monodromies of one-parameter degenerations of complex algebraic varieties
where the bound of the size comes from the monodromy theorem. In case the
general fibers are smooth and compact, the proof calculates some part of the
weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In
the singular case, we can prove a similar formula for the monodromy on the
cohomology with compact supports, but not on the usual cohomology. We also show
that the number can really depend on the position of singular points in the
embedded resolution even in the isolated singularity case, and hence there are
no simple combinatorial formulas using the embedded resolution in general.Comment: 23 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
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