Implicitly defined harmonic PHH submersions

The concept of harmonic pseudo-horizontally homothetic submersion, alias harmonic PHH submersion, expresses the more geometric notion of smooth family of minimal submanifolds indexed by a K?hler parameter manifold. We give results that allow to construct harmonic PHH submersions by solving some implicit equations and we apply this method to the case of sphere bundles. Received: 3 September 1998 / Accepted: 19 March 1999     

On the vanishing of higher syzygies of curves. II

Mathematische Zeitschrift -     

Boundedness for Some Rationally Connected Threefolds in ℙ6

We prove boundedness of rationally-connected threefolds in ?⁶ under some extra-assumptions.     

Enriques diagrams and log-canonical thresholds of curves on smooth surfaces

It is shown that, on a smooth surface, the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.     

A Note on vector bundles on Hirzebruch surfaces

In literature, two basic construction methods have been used to study vector bundles on a Hirzebruch surface. On the one hand, we have Serre?s method and elementary modifications, describing rank-2 bundles as extensions in a canonical way (Brînz?nescu and Stoia, 1984 [4], [5], Brînz?nescu, 1996 [6], Brosius, 1983 [7], Friedman, 1998 [9]), and on the other hand, we have a Beilinson-type spectral sequence (Buchdahl, 1987 [8]). Morally, the Beilinson spectral sequence indicates how to recover a bundle from the cohomology of its twists and from some sheaf morphisms (the differentials of the sequence). The aim of this Note is to show that the canonical extension of a rank-2 bundle can be deduced from the Beilinson spectral sequence of a suitable twist, called the normalization. In the final part we give a cohomological criterion for a topologically trivial vector bundle on a Hirzebruch surface to be trivial. To emphasize the relations and the differences between these two construction methods mentioned above, two different proofs are given.     

Une Note sur les fibrés holomorphes non-filtrables

We prove that any non-filtrable holomorphic rank-2 vector bundle on a non-Kähler elliptic surface is an elementary modification of a direct image of a line bundle by a double covering of the surface. To cite this article: M. Aprodu, M. Toma, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Green–Lazarsfeld's conjecture for generic curves of large gonality

We use Green's canonical syzygy conjecture for generic curves to prove that the Green–Lazarsfeld gonality conjecture holds for generic curves of genus g, and gonality d, if g/3<d<[g/2]+2. To cite this article: M. Aprodu, C. Voisin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Ulrich line bundles on Enriques surfaces with a polarization of degree four

In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles.