ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14 Ronconi, M. Cristina, University of Padua, Italy Algebraic threefolds with q_{1} = q_{2} = p_{g} = 0, P_{2i} = 1 and P_{2i+1} = 0 Research on threefolds with properties similar to those of Enriques surfaces (p_{g} = 0, P_{2i} = 1, P_{2i+1} = 0) has been carried out by various authors, including L. Godeaux and K. Ueno. In addition to the already-mentioned birational invariants, Enriques surfaces S have h^{1}(S, O_{S}) = 0, 2K_{S} ≡ 0 and contain a pencil of elliptic curves, but not all of these properties can coexist on a threefold X because, e. g., if 2K_{X} ≡ 0 then q_{1} = h^{1}(X, O_{X}) > 0. Results of a research currently under way on threefolds with p_{g} = q_{1} = q_{2} = 0, P_{2i} = 1, P_{2i+1} = 0 will be presented: the purpose is to prove the existence of threefolds of this type, obtained as a nonsingular model of a degree six hypersurface V_{6} ⊂ ℙ_{ℂ}^{4} containing a pencil of surfaces with Kodaira dimension = 0 or a net of elliptic curves.