The universal rank-(n-1) bundle on G(1,n) restricted to subvarieties

We classify those smooth (n-1)-folds in G(1,n) for which the restriction of the rank-(n-1) universal bundle has more than n+1 independent sections. As an aplication, we classify also those (n-1)-folds for which that bundle splits.Comment: 11 pages, plain te

A home-made Hartshorne-Serre correspondence

We provide an elementary proof of the Hartshorne-Serre correspondence for constructing vector bundles from local complete intersection subschemes of codimension two. This will be done, as in the correspondence of hypersurfaces and line bundles, by patching together local determinantal equations in order to produce sections of a vector bundle.Comment: 19 page

Vector bundles on G(1,4) without intermediate cohomology

We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector bundles from their cohomology.Comment: 12 page

A focus on focal surfaces

We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal surface (degree, class, class of its hyperplane section, sectional genus and degrees of the nodal and cuspidal curve). We study in particular the congruences of chords to a smooth curve and the congruences of bitangents or flexes to a smooth surface. We find that they possess unexpected components in their focal surface, and conjecture that they are the only ones with this property.Comment: Plain TeX, 33 pages with no figure

Optical spin control in nanocrystalline magnetic nanoswitches

We investigate the optical properties of (Cd,Mn)Te quantum dots (QDs) by looking at the excitons as a function of the Mn impurities positions and their magnetic alignments. When doped with two Mn impurities, the Mn spins, aligned initially antiparallel in the ground state, have lower energy in the parallel configuration for the optically active spin-up exciton. Hence, the photoexcitation of the QD ground state with antiparallel Mn spins induces one of them to flip and they align parallel. This suggests that (Cd,Mn)Te QDs are suitable for spin-based operations handled by light

On the variety parametrizing completely decomposable polynomials

The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree $d$ in $n+1$ variables on an algebraically closed field, called \Split_{d}(\PP n), with the Grassmannian of $n-1$ dimensional projective subspaces of \PP {n+d-1}. We compute the dimension of some secant varieties to \Split_{d}(\PP n) and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to \GG (n-1, n+d-1). Moreover by using an invariant embedding of the Veronse variety into the Pl\"ucker space, then we are able to compute the intersection of \GG (n-1, n+d-1) with \Split_{d}(\PP n), some of its secant variety, the tangential variety and the second osculating space to the Veronese variety.Comment: 30 page