74 research outputs found
Basic nets in the projective plane
The notion of basic net (called also basic polyhedron) on plays a
central role in Conway's approach to enumeration of knots and links in .
Drobotukhina applied this approach for links in using basic nets on
. By a result of Nakamoto, all basic nets on can be obtained from a
very explicit family of minimal basic nets (the nets , ,
in Conway's notation) by two local transformations. We prove a similar result
for basic nets in .
We prove also that a graph on is uniquely determined by its pull-back
on (the proof is based on Lefschetz fix point theorem).Comment: 14 pages, 15 figure
On the number of singular points of plane curves
This is an extended, renovated and updated report on a joint work which the
second named author presented at the Conference on Algebraic Geometry held at
Saitama University, 15-17 of March, 1995. The main result is an inequality for
the numerical type of singularities of a plane curve, which involves the degree
of the curve, the multiplicities and the Milnor numbers of its singular points.
It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality
due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the
authors independently in the particular case of an irreducible cuspidal curve
at 1992. Our proof is based on the localization, the local Zariski--Fujita
decomposition and uses a graph discriminant calculus. The key point is a local
analog of the BMY-inequality for a plane curve germ. As a corollary, a
boundedness criterium for a family of plane curves has been obtained. Another
application of our methods is the following fact: a rigid rational cuspidal
plane curve cannot have more than 9 cusps.Comment: LaTeX, 24 pages with 3 figures, author-supplied DVI file available at
http://www.math.duke.edu/preprints/95-00.dv
The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves
The number of topologically different plane real algebraic curves of a given
degree has the form . We determine the best available
upper bound for the constant . This bound follows from Arnold inequalities
on the number of empty ovals. To evaluate its rate we show its equivalence with
the rate of growth of the number of trees half of whose vertices are leaves and
evaluate the latter rate.Comment: 13 pages, 3 figure
Plane curves with a big fundamental group of the complement
Let C \s \pr^2 be an irreducible plane curve whose dual C^* \s \pr^{2*}
is an immersed curve which is neither a conic nor a nodal cubic. The main
result states that the Poincar\'e group \pi_1(\pr^2 \se C) contains a free
group with two generators. If the geometric genus of is at least 2,
then a subgroup of can be mapped epimorphically onto the fundamental group
of the normalization of , and the result follows. To handle the cases
, we construct universal families of immersed plane curves and their
Picard bundles. This allows us to reduce the consideration to the case of
Pl\"ucker curves. Such a curve can be regarded as a plane section of the
corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying
Zariski--Lefschetz type arguments we deduce the result from `the bigness' of
the -th braid group of the Riemann surface of .Comment: 23 pages LaTeX. A revised version. The unnecessary restriction of the previous version has been removed, and the main result has
taken its final for
- …