ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14 Algebraic geometry Conte, Alberto, Universitá di Torino, Italia On the unirationality of quartic hypersurfaces It is well known that the general quartic hypersurface V_{r−1}(4) ⊆ P^{r}, defined over any field k, is unirational for r ≥ 6. This follows, for r ≥ 7, from a general result of C. Ciliberto, whilst, for r = 6, a direct proof was given by U. Morin. We show how a much simpler proof than Morin’s can be given using a theorem of B. Segre and give hints on how one could possibly attack the open cases r = 4, 5. In particular, we show that quartic fourfolds containing a plane (or a Hirzebruch surface F_{n}) are unirational. These results are joint work with J. P. Murre and A. Verra.