Upper bounds for errors of estimators in a problem of nonparametric regression: the adaptive case and the case of unknown measure ρX

The topological type of the real part of the Fano variety parametrizing the set of lines on a nonsingular real hypersurface of degree three in a five-dimensional projective space is evaluated provided that the hypersurface belongs to a special rigid projective class. In the paper by Finashin and Kharlamov on the rigid projective classification of real four-dimensional cubics, this class is said to be irregular. The results of the author of the present paper from the article devoted to the equivariant topological classification of the Fano varieties of real cubic fourfolds are also used.     

Equivariant topological classification of the Fano varieties of real four-dimensional cubics

Mathematical Notes - The equivariant topological type of the Fano variety parametrizing the set of lines on a nonsingular real hypersurface of degree three in a five-dimensional projective space is...     

Uniform Zariski’s theorem on fundamental groups

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates an isomorphism of the fundamental groups of the complements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution. The research was partially supported by an NSA grant.     

Existenzbedingungen für Dualitäten projektiver Ebenen

In a projective plane a permutation of the set of all points and lines is constructed, using only the operations join and meet. Under certain conditions (identities in a coordinatizing ternary field; special cases of Desargues- and Pappos-theorem) this permutation is a duality. For a topological projective plane this duality proves, that point space and line space of the plane are homeomorphic.     

Sylvester–Gallai Theorems for Complex Numbers and Quaternions

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a deep inequality of Hirzebruch. We give an elementary proof of this result, and then extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.     

Real hypersurfaces in quaternionic projective space

Summary This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.     

Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP ⁿ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP ⁿ is harmonic, and show that the answer is in negative.     

Pseudosymmetric Lightlike Hypersurfaces in Indefinite Sasakian Space Forms

We study pseudosymmetric lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field. We obtain sufficient conditions for a lightlike hypersurface to be pseudosymmetric, pseudoparallel and Ricci-pseudosymmetric in an indefinite Sasakian space form. We also find certain conditions for a pseudosymmetric lightlike hypersurface of an indefinite Sasakian space form to be totally geodesic and check the effect of Weyl projective pseudosymmetry conditions on the geometry of a lightlike hypersurface of an indefinite Sasakian space form. Moreover we give some physical interpretations of pseudo-symmetry conditions.     

On tangential deformations of homogeneous polynomials

The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is indeed trivial, and show that the answer is no in a general situation. We also give a characterization of tangentially smoothable hypersurfaces with isolated singularities. Our results have applications in the local study of variations of projective hypersurfaces, complementing the global versions given by J. Carlson and P. Griffiths, R. Donagi and the author, and in the study of isotrivial linear systems on the projective space, showing that a general divisor does not belong to an isotrivial linear system of positive dimension.     

Homotopy classification of projective convex sets

A subset of projective space is called convex if its intersection with every line is connected. The complement of a projective convex set is again convex. We prove that for any projective convex set there exists a pair of complementary projective subspaces, one contained in the convex set and the other in its complement. This yields their classification up to homotopy.     

Blowing up points and embedding flat stable planes in the nonorientable compact surface of genus one

We show that the point set of every flat stable plane embeds in the point set of the real projective plane. Connectedness of lines or of the point space is not assumed. We give two largely independent proofs; the first one is more conceptual, while the second one is more direct, and shorter. The first proof uses a new construction called blowing up a point, i.e., replacing it with its line pencil; this amounts to adding a cross cap. This construction seems to be of interest in its own right.     

Low-degree rational curves on hypersurfaces in projective spaces and their fan degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.     

Constructions of Hjelmslev planes

We present a new construction method for strongly n-uniform Hjelmslev planes which gives greater control in selecting the point and line neighborhood structures of the constructed planes. As a consequence, we are able to complete the proof of the Drake-Törner Theorem which asserts: the spectrum of invariant pairs for the class of finite, regular, minimally uniform projective Hjelmslev planes is the set of all Lenz-pairs.     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

The number of reducible space curves over a finite field

“Most” hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.     

复射影空间中具有常数量曲率的实超曲面

将球面上常数量曲率超曲面推广到复射影空间中,得到此类实超曲面的某些积分不等式.     

Projective Minkowski set

The Minkowski set or the central symmetry set (CSS) of a smooth curve Γ on the affine plane is the envelope of chords connecting pairs of points such that the tangents to Γ at them are parallel. Singularities of CSS are of interest, in particular, for applications (for example, in computer graphics). A generalization of the Minkowski set is considered in the paper, namely, the projective Minkowski set with respect to a line on the plane; in the case of general position, we describe its singularities and the bifurcation set of lines corresponding to lines defining the projective Minkowski set having singularities being more degenerate than those of the Minkowski set for a generic line.     

On the tangent spaces of Chow groups of certain projective hypersurfaces

In this paper, the Chow groups of projective hypersurfaces are studied. We will prove that if the degree of the hypersurface is sufficiently high, its Chow group is ``small' in the sense that its formal tangent space vanishes. Then, we will give an example in which the formal tangent space is infinite dimensional.

     

Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity

The results we obtain in this article concern the hyperbolicity of very generic hypersurfaces in the 3-dimensional projective space: we show that the Kobayashi conjecture is true in this setting, as long as the degree of the hypersurface is greater than 18.     

Focal loci of algebraic varieties i

The focal locus ∑_x of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x ∈X and a circle with centre O passing through x which osculates X inx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle N_x and the projective ambient space (∈ sends the normal vector O - x to its endpoint O), and in this paper we address two general problems:.

1)Characterize the"degenerate"case where the focal locus is not a hyper surface.

2)Calculate, in the case where ∑x is a hypersurface, its degree (with multiplicity).