476 research outputs found
Twisted Alexander polynomials of Plane Algebraic Curves
We consider the Alexander polynomial of a plane algebraic curve twisted by a
linear representation. We show that it divides the product of the polynomials
of the singularity links, for unitary representations. Moreover, their quotient
is given by the determinant of its Blanchfield intersection form. Specializing
in the classical case, this gives a geometrical interpretation of Libgober's
divisibility Theorem. We calculate twisted polynomials for some algebraic
curves and show how they can detect Zariski pairs of equivalent Alexander
polynomials and that they are sensitive to nodal degenerations.Comment: 16 pages, no figure
Effective invariants of braid monodromy and topology of plane curves
In this paper we construct effective invariants for braid monodromy of affine
curves. We also prove that, for some curves, braid monodromy determines their
topology. We apply this result to find a pair of curves with conjugate
equations in a number field but which do not admit any orientation-preserving
homeomorphism.Comment: 26 pages, two EPS figures, LaTe
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
The Max Noether Fundamental Theorem is Combinatorial
In the present paper we give a reformulation of the Noether Fundamental
Theorem for the special case where the three curves involved have the same
degree. In this reformulation, the local Noether's Conditions are weakened. To
do so we introduce the concept of Abstract Curve Combinatorics (ACC) which will
be, in the context of plane curves, the analogue of matroids for hyperplane
arrangements
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