Certain relative quadric hypersurfaces in 3-dimensional projective space bundles over a projective curve

We classify the certain type of relative quadric hypersurfaces of 3-dimensional projective space bundles over a projective line or an elliptic curve whose fiber is the direct product of 2 projective lines.     

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface \(X \subset \mathbf {CP}^{n}\). Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface \(X\): we construct a configuration of monotone Lagrangian spheres in \(X\), and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of \(X\) (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for \(X\). We also explain how to extract non-trivial information about Gromov–Witten invariants of \(X\) from its Fukaya category.     

On the connectedness of the scheme of multisecants to a projective curve

Here we study the connectedness of the scheme of multisecant linear spaces to a curveC P ⁿ . In particular we prove the connectedness of the scheme of trisecant lines of a smoothC P ³ with non-special hyperplane section.     

On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

On the order structure in the line geometry of a projective space

A definition is made of an ordered Grassmann space and every ordered Grassmann space is shown to be isomorphic to the Grassmann space related to an ordered projective space.Dedicated to Professor W. Benz on the occasion of his 60^th birthday     

The point and line space of a compact projective plane are homeomorphic

The local Hopf bundles at the points and lines of a compact projective plane with manifold lines are all weakly equivalent. In particular, the point space and the line space of such a projective plane are homeomorphic. This is a consequence of the following topological result. Theorem. Let $\xi,\xi'$ be orientable topological $\mathbb{R}^n$ -bundles over an n-dimensional CW-complex. If $\xi$ and $\xi'$ are fibre homotopy equivalent and stably equivalent, then $\xi$ and $\xi'$ are equivalent. Received: 10 December 1998 / Accepted: 18 May 1999     

On the projective differential geometry of a non-holonome surface in ordinary space

Sunto. The paper deals with many projective properties of the neighbourhoods of the third,4th, 5th order of a point on a non-holonome surfaceV ₃ ² inS ₃. Chiefly two remarkable projective correspondences are studied, between the tangent plane and the bundle of directions, with his contact point as centre. Many generalizations are obtained of geometrical loci (so as lines, planes, quadrics) projectively associated with the neighbourhoods of a point on an ordinary (holonome) surface. The last sections are concerned with the extension to theV ₃ ² of the quadric ofMoutard and theSegre's correspondence.      

On the Bergman metric of pseudoconvex domains in a complex projective space

We prove a localization principle of the Bergman kernel form and metric for pseudoconvex domains in the complex projective space. An estimate of the Bergman distance is also given.

     

On the classification of fuzzy projective planes of fuzzy 3-dimensional projective space

In this work, the classifications of fuzzy 3-dimensional vector spaces of fuzzy 4-dimensional vector space and fuzzy projective planes of fuzzy 3-dimensional projective space from fuzzy 4-dimensional vector space are given.