16 research outputs found
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
DEFORMATION CLASSES OF REAL FOUR-DIMENSIONAL CUBIC HYPERSURFACES
We study real nonsingular cubic hypersurfaces X subset of P-5 up to deformation equivalence combined with projective equivalence and prove that, they, are classified by the conjugacy classes of involutions induced by the complex conjugation in H-4(X). Moreover, we provide a graph Gamma(K4) whose vertices represent the equivalence classes of such cubics and whose edges represent their adjacency. It turns out that the graph Gamma(K4) essentially coincides with the graph Gamma(K3) characterizing a certain adjacency of real nonpolarized K3-surfaces
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
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
Two kinds of real lines on real del Pezzo surfaces of degree 1
We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, Pin(-)-structure on the real locus of the surface. We prove that this splitting is invariant under real automorphisms and real deformations of the surface, and that the difference between the total numbers of hyperbolic and elliptic lines is always equal to 16
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Abundance of 3-planes on real projective hypersurfaces
We show that a generic real projective n-dimensional hypersurface of
odd degree d , such that 4(n - 2) = (d + 3 3), contains "many" real 3-planes, namely,
in the logarithmic scale their number has the same rate of growth, d3 log d, as the
number of complex 3-planes. This estimate is based on the interpretation of a suitable
signed count of the 3-planes as the Euler number of an appropriate bundle