Projective surfaces with bi-elliptic hyperplane sections

We study projective surfaces X which have a bi-elliptic curve (i.e. 2∶1 covering of an elliptic curve) among their hyperplane sections . We give a complete characterization of those surfaces when their degree d is d≥17 (only conic bundles and scrolls if d≥19, three possible exception otherwise) and when d≤8. A conjecture is given for the remaining cases. The main tool we use is the study of the adjunction mapping on X.     

Three-dimensional manifolds with hyperplane sections which are ruled surfaces

A nonsingular 3-dimensional algebraic manifold over the field of complex numbers, whose general hyperplane section is a ruled surface with irregularity q>0, is birationally equivalent to the direct product of the projective plane and a nonsingular curve of genus q.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 821–826, December, 1973.     

The Nash modification and hyperplane sections on surfaces

We prove that the planar components of the tangent cone of a complex analytic surface at a point correspond to the base points of hyperplane sections by the Nash modification. This correspondence is then used to characterize domination relations between the normalized Nash modification and the normalized blow-up of a point.     

On Riemann surfaces with non-unique cyclic trigonal morphism

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called cyclic trigonal Riemann surface. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus greater or equal to 5. Using the characterization of cyclic trigonality by Fuchsian groups given in [3], we obtain the Riemann surfaces of low genus with non-unique trigonal morphisms. Partially supported by BFM2002-4801. Partially supported by the Swedish Research Council (VR)     

On non-isotropic harmonic maps of surfaces into complex projective spaces

It is proved by purely algebraic method that weakly conformai, conformai andA _z ³ = 0 are mutually equivalent if ϕ :Ω→ℂP ⁿ is a non-isotropic harmonic map and the harmonic maps with isotropy order ≥3 are uniquely determined by a system of ordinary differential equations. A method is given, by which the isotropy orders of non-isotropic harmonic maps can be computed.     

Meromorphic singular foliations on complex projective surfaces

Here we show that many numerical computations and bounds on the degrees of the algebraic leaves for singular meromorphic foliations on ² may be extended to large classes of foliations and complex projective surfaces.The author was partially supported by MURST and GNSAGA of CNR (Italy)     

Non degenerate projective curves with very degenerate hyperplane section

In this paper, we study the Hilbert scheme of non degenerate locally Cohen- Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component H_n,d,g of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of H_n,d,g.All the authors were partially supported by Acción Integrada Italia-España, HI2000-0091, and by the Italian counterpart of the project.     

Subadjoint ideals and hyperplane sections

We study the behaviour of the notion of ``sub-adjoint ideal to a projective variety" with respect to general hyperplane sections. As an application we show that the two classical definitions of sub-adjoint hypersurface given respectively by Enriques and Zariski are equivalent.

     

Flat slant surfaces in complex projective and complex hyperbolic planes

For surfaces in complex space forms with almost complex structure J, flat surfaces are the simplest ones from intrinsic point of view. From J-action point of view, the most natural surfaces are slant surfaces. The classification of flat slant surfaces in C² was done in [2]. In this paper we apply a result of [5] to study flat slant surfaces in CP ² and CH ². We prove that, for any θ, there exist infinitely many flat θ-slant surfaces in CP ² and CH ². And there does not exist flat half-minimal proper slant surface in CP ² and in CH ².     

A characterization of integral curves with Gorenstein hyperplane sections

We classify a reduced, irreducible and non-degenerate curve such that its general hyperplane section is arithmetically Gorenstein, but itself is not. These curves are contained in surface scrolls and are closely related to Castelnuovo theory on curves in projective space.

     

Some remarks on varieties with projectively isomorphic hyperplane sections

We study here the projective varieties with the property that there exists a projective isomorphism between two of their generic hyperplane sections. The case of surfaces had already been studied by Fubini and Fano in the 1920s. The latter gave the list of all (possibly signular) surfaces with projectively isomorphic hyperplane sections. The proof, however, was essentially wrong. By means of a different approach, we are able to supply a proof of Fano's claims. Moreover, we show some general properties of varieties with projectively isomorphic hyperplane sections: they have uniruled hyperplane sections and are related to varieties with small dual varieties. In particular we are able to conclude that threefolds with projectively isomorphic hyperplane sections either have rational sections or are P²-bundles over a curve.A member of GNSAGA of CNR.     

On projective embeddings of generic rational surfaces

Let R be a commutative ring and let ?(σ) be a Gabriel filter of R such that R is σ-noetherian. We discuss the decomposition of the σ ¹-torsion submodule of a σ-torsionfree R-module and characterize the σ ^l-injectivity of σ-closed R-modules through the σ _m-injectivity of modules over noetherian local rings (S, m). As an application, we obtain new criteria to determine injectivity of modules over noetherian rings, of finite Krull dimension, and Krull domains.