Multiprojective Spaces Research Articles

Foliations of multiprojective spaces and a conjecture of Bernstein and Lunts

We use foliations of multiprojective spaces defined by Hamiltonian functions on the underlying affine space to prove the three dimensional case of a conjecture of Bernstein and Lunts, according to which the symbol of a generic first-order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle.

A few splitting criteria for vector bundles

We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson’s type spectral sequence generalized by Costa and Miro-Roig.

The stability of certain vector bundles on multi-projective spaces

The stability of certain vector bundles on multi-projective spaces

Subschemes of multi-projective spaces and the generators of their multi-homogeneous ideal

Subschemes of multi-projective spaces and the generators of their multi-homogeneous ideal

Cohomological characterization of vector bundles on multiprojective spaces

We show that Horrocks criterion for the splitting of vector bundles on P n can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool we use the theory of n-blocks and Beilinson type spectral sequences. Cohomological characterizations of vector bundles are also showed.

The regularity of points in multi-projective spaces

Let I=℘ 1 m 1 ∩⋯∩℘ s m s be the defining ideal of a scheme of fat points in P n 1 ×⋯× P n k with support in generic position. When all the m i 's are 1, we explicitly calculate the Castelnuovo–Mumford regularity of I. In general, if at least one m i ⩾2, we give an upper bound for the regularity of I, which extends a result of Catalisano, Trung and Valla.

Dynamique des applications rationnelles des espaces multiprojectifs

We study the dynamics of rational mappings f of C k by compactifying them in multiprojective spaces P n1 P ns. We focus on maps of the surface P 1 P 1 . We follow the ap- proach of (Si 99) and associate to any algebraically stablef an in- variant positive closedN1; 1O current. We then consider the exis- tence of anf invariant measure using the theory of pluripositive currents, and relates it to the measure of Russsakovskii-ShiVman describing the distribution of preimages of points. Our point of view enables us to treat new classes of examples: we consider in particular polynomial skew products with varying degrees, and birational polynomial mappings ofC 2 . We also describe the com- pact convex set off invariant currents for monomial and bira- tional maps of C 2 .