We generalize Griffiths’ theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface, using the local Bernstein-Sato polynomials, the V -filtration of Kashiwara and Malgrange along the hypersurface and the Brieskorn module of the global defining equation of the hypersurface.

We complete the study of birational geometry of Fano fiber spaces π: V → P 1 , the fiber of which is a Fano double hypersurface of index 1. For each family of these varieties we either prove birational rigidity or produce explicitly non-trivial structures of Fano fiber spaces. A new linear method of studying movable systems on Fano fiber spaces V /P 1 is developed. Bibliography: 17 titles.

For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix...

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of the MacRae’s invariant of SymA(I)ν is optimal. Then, we show that the extraneous factor that may appear in the process splits into a product a linear forms in...

α∈Q nαt , nα in Z, which is constructed using the action of the monodromy on the mixed Hodge structure on the cohomology of the Milnor fiber at x. When f has an isolated singularity at x, all nα are in N, and the exponents of f , counted with multiplicity nα, are exactly the rational numbers α with nα not zero. Let us assume now that the singular locus of f is a curve Γ, having r local componen...

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, set of solutions is called the curvature diagram or W-diagram surface. Making use notion geometric linear momentum plane curve, we propose new approach to study rotational in Euclidean 3-space. Our contribution consists reducing any type condition on surface first-order differential equation generatrix curve...