Narrow Results

Determining the weight distributions of the projective Reed–Muller codes is a very hard problem and has been studied extensively in the literature. In this paper, we provide an alternative proof of the second weight of the projective Reed–Muller codes ${\mathrm{\text{PRM}}}_q(d,m)$ where $m\ge 3$ and $3\le d\le \frac{q+3}{2}$. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on $d$. Furthermore, we compute the second weight of ${\mathrm{\text{PRM}}}_q(d,2)$ for $3\le d\le q$. Furthermore, we give an upper bound for the third weight of ${\mathrm{\text{PRM}}}_q(d,2)$.

We show that the usual sufficient criterion for a very general hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to quasi-smooth hypersurfaces in complete simplicial toric varieties. This sufficient condition always holds for very general K3 surfaces embedded in Fano toric 3-folds.

Using the geometry of geodesics, we discuss the global aspects of complete real hypersurfaces in hyperbolic spaces of constant holomorphic sectional curvature over any division algebra 𝕂. Our assumption is that the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. Note that we do not assume constancy of the principal curvatures. Under this assumption, we give a complete global classification of such hypersurfaces. Since the argument is purely geometric, we need not vary the argument for different base algebras. The foliations of with totally geodesic leaves called 𝕂-lines play an important role.

The Noether–Lefschetz theorem asserts that any curve in a very general surface $X$ in ${\mathbb{P}}^3$ of degree $d\ge 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d\ge 4$, a very general surface in a simplicial toric threefold ${\mathbb{P}}_{\mathrm{\Sigma }}$ (with orbifold singularities) has the same Picard number as ${\mathbb{P}}_{\mathrm{\Sigma }}$. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in ${\mathbb{P}}_{\mathrm{\Sigma }}$ in a linear system of a Cartier ample divisor with respect to a $(-1)$-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ($Q$-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson’s $Q$-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

In this paper, we study Euler's theorem for semi-Euclidean hypersurfaces in the semi-Euclidean spaces . We obtain an analog of the well-known Euler's theorem for semi-Euclidean hypersurfaces in the semi-Euclidean spaces . Then we give corollaries of Euler's theorem concerning conjugate and asymptotic directions. After that, we express Euler's theorem and its corollaries for hypersurfaces in the Euclidean space 𝔼^m in the case n = m - 1, v = 0. In addition, we give the well-known Euler's theorem and its corollaries for surfaces in the case n = 2, v = 0, for Lorentz surfaces in the case n = 2, v = 1 and for hypersurfaces in Lorentz spaces in the case n = m - 1, v = 1.

The hypersurface of a space is one of the most important objects in a space. Many authors studied the various geometric aspects of hypersurfaces in a space form. The notion of conformal flatness is one of the most primitive concepts in differential geometry. Again, conformally flat space is a proper generalization of a space form. In this paper, we study the geometry of hypersurfaces in a conformally flat manifold. Then we have investigated some sufficient conditions imposed on the shape operator for which the hypersurface satisfies various pseudosymmetric-type conditions imposed on its conformal curvature tensor.

Nous démontrons ici la conjecture de Batyrev et Manin pour le nombre de points de hauteur bornée de certaines hypersurfaces de l'espace triprojectif de tridegré (1, 1, 1). La constante intervenant dans le résultat final est celle conjecturée par Peyre. La méthode utilisée est inspirée de celle développée par Schindler pour traiter le cas des hypersurfaces des espaces biprojectifs. Celle-ci est essentiellement basée sur la méthode du cercle de Hardy–Littlewood.

We prove the Batyrev–Manin conjecture for the number of points of bounded height on some smooth hypersurfaces of the triprojective space of tridegree (1, 1, 1). The constant appearing in the final result is the one conjectured by Peyre. The method used is the one developed by Schindler to study the case of hypersurfaces of biprojective spaces. It is essentially based on the Hardy–Littlewood method.

In this paper, we generalize and extend some recent unicity theorems for meromorphic mappings of a complete Kähler manifold into ${\mathbb{P}}^n(C)$ sharing few hyperplanes as well as hypersurfaces in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number do not need to be counted. Further, the assumption of nondegeneracy of the mappings is omitted in some of our results.

This is a half survey on the classical results of extrinsic differential geometry of hypersurfaces in Euclidean space from the view point of Lagrangian or Legendrian singularity theory. Many results in this paper have been already obtained in some articles. However, we can discover some new information of geometric properties of hypersurfaces from this point of view.

The following sections are included:

• Conformally flat manifolds

• Conformally flat submanifolds

• Three dimensional conformally flat hypersurfaces

• References

We prove that a hypersurface of a space form with almost constant mean curvature and almost constant scalar curvature is close to a geodesic sphere. In the case of Euclidean space, we deduce new characterizations of geodesic spheres.