We consider regular Calabi-Yau hypersurfaces in N -dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere S whose generic fibers are tori T. Also for certain one-parameter families of such hypersurfaces we show that the monodromy transformation is induced by a translation of the T fibration...

In this paper we give some geometric characterizations of locally convex hypersurfaces. In particular, we prove that for a given locally convex hypersurface M with boundary, there exists r > 0 depending only on the diameter of M and the principal curvatures of M on its boundary, such that the r-neighbourhood of any given point on M is convex. As an application we prove an existence theorem for ...

We present a new method for implicitization of parametric curves, surfaces and hypersurfaces using essentially numerical linear algebra. The method is applicable for polynomial, rational as well as trigonometric parametric representations. The method can also handle monoparametric families of parametric curves, surfaces and hypersurfaces with a small additional amount of human interaction. We i...

LetX be the base locus of a linear system L of hypersurfaces in Pr(C). In this paper it is showed that the existence of linear syzygies for the ideal of X has strong consequences on the fibres of the rational map associated to L. The case of hyperquadrics is particularly addressed. The results are applied to the study of rational maps and to the Perazzo’s map for cubic hypersurfaces.

We study some Weingarten spacelike hypersurfaces in a de Sitter space S 1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product H(1−coth ̺)× S(1 − tanh ̺), 1 < k < n − 1 in S 1 (1), the hyperbolic cylinder H(1 − coth ̺) × S(1 − tanh ̺) or spherical cylinder H(1 − coth ̺)×...

The paper deals with singularities of nonconfluent hypergeometric functions in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of amoebas and the Newton polytopes of their defining polynomials. In particular, we show that allA-discriminantal hypersurfaces (in the sense of...

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed ...

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curva...

Our object is to extend earlier work [D1] on singular hypersurfaces defined over an algebraic number field to singular hypersurfaces defined over function fields in characteristic zero. A key role will be played by the results of Bertolin [B1] which in turn is based upon the Transfer Theorem of André–Baldassarri–Chiarellotto [DGS, Theorem VI 3.2]. Let h(A, x) be the generic form of degree d in ...

By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker...