2,691 research outputs found
On the deformation chirality of real cubic fourfolds
According to our previous results, the conjugacy class of the involution
induced by the complex conjugation in the homology of a real non-singular cubic
fourfold determines the fourfold up to projective equivalence and deformation.
Here, we show how to eliminate the projective equivalence and to obtain a pure
deformation classification, that is how to respond to the chirality question:
which cubics are not deformation equivalent to their image under a mirror
reflection. We provide an arithmetical criterion of chirality, in terms of the
eigen-sublattices of the complex conjugation involution in homology, and show
how this criterion can be effectively applied taking as examples -cubics
(that is those for which the real locus has the richest topology) and
-cubics (the next case with respect to complexity of the real locus). It
happens that there is one chiral class of -cubics and three chiral classes
of -cubics, contrary to two achiral classes of -cubics and three
achiral classes of -cubics.Comment: 25 pages, 8 figure
Deformation classification of real non-singular cubic threefolds with a marked line
We prove that the space of pairs formed by a real non-singular cubic hypersurface with a real line has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface formed by real lines on . For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of characterizes completely the component
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
On braid monodromy factorizations
We introduce and develop a language of semigroups over the braid groups for a
study of braid monodromy factorizations (bmf's) of plane algebraic curves and
other related objects. As an application we give a new proof of Orevkov's
theorem on realization of a bmf over a disc by algebraic curves and show that
the complexity of such a realization can not be bounded in terms of the types
of the factors of the bmf. Besides, we prove that the type of a bmf is
distinguishing Hurwitz curves with singularities of inseparable types up to
-isotopy and -holomorphic cuspidal curves in \C P^2 up to symplectic
isotopy.Comment: 52 pages, AMS-Te
Topology of real cubic fourfolds
A solution to the problem of topological classification of real cubic
fourfolds is presented. It is shown that the real locus of a real non-singular
cubic fourfold is obtained from a projective 4-space either by adding several
trivial one- and two-handles, or by adding a spherical connected component.Comment: 28 pages, 6 figure
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