On the pluricanonical map of threefolds of general type

Let be a smooth minimal threefold of general type and let be an integer . Assume that the image of the pluricanonical map of is a curve. Then a simple computation shows that is necessarily or . When with a numerical condition or when , we obtain two inequalities and , where is the irregularity of and is the Euler characteristic of .

     

Automorphisms of threefolds of general type acting trivially in cohomology

Geometriae Dedicata - Let X be a minimal projective threefold of general type over $$\mathbb {C}$$ with only Gorenstein quotient singularities, and let $$\mathrm {Aut}_{\mathbb {Q}}(X)$$ be the...     

A note on threefolds with a hyperplane section of general type

This paper deals with polarized pairs , where is a nonsingular projective threefold and is a very ample line bundle on it, such that for one smooth member  | |, one has (Â)=2. A large class of pairs whose adjoint line bundle is nef and big was indirectedly studied by Beltrametti and co-workers. We add some more information, both in this general case and also when the adjoint line bundle fails to be nef and big.     

On the projectivity of threefolds

Let be a smooth complete three-dimensional algebraic variety (defined over an algebraically closed field ). We show that is projective if it contains a divisor which is positive on the cone of effective curves.

     

Regular threefolds of general type with p g =0 and large bigenus

The known examples of nonsingular complex projective threefolds of general type with geometric genus p _g =0 and irregularities q ₁=q ₂=0 have bigenus P ₂≤6. In the present paper nonsingular threefolds with p _g =q ₁=q ₂=0 and P ₂=7 or P ₂=8 are constructed.     

On the classification of Hilbert modular threefolds

Letk be a totally real number field with ring of integersO _k . The Hilbert modular variety overk is a desingularization of the (natural) compactification of PSL₂(O _k )∖H ^k . The purpose of this paper is to present specific numerical bounds on the size of the discriminantd _k of a cubic fieldk with Hilbert modular variety of particular classifications. specifically, it is shown that ifd _k>2.12×10⁷, then the Hilbert modular variety overk is not rational and further, ifd _k>2.77×10⁸, then Hilbert modular variety overk is of general type. This material is based on work supported by the National Science Foundation under Grant No. DMS-9008689     

On the Kodaira dimension of minimal threefolds

Dedicated to Professor M. Nagata on his sixtieth birthday     

On compactifiable strongly pseudoconvex threefolds

In contrast with the 2-dimensional case, we would like to exhibit here an example of a compactifiable strongly pseudoconvex threefold X such that a) the analytic Kodaira dimension κ(X)=3 and b) its exceptional subvariety S is projective algebraic; yet X is not quasi-projective.     

On threefolds covered by lines

A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, ifX is such a threefold, let Σ denote the Fano scheme of lines onX and μ the number of lines contained inX and passing through a general point ofX. Assume that Σ is generically reduced. Then μ ≤ 6. Moreover,X is birationally a scroll over a surface (μ = 1), orX is a quadric bundle, orX belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with −K _X = 2H _X . This work has been done in the framework of the activities of EAGER. Both authors have been supported by funds of MURST, Progetto Geometria Algebrica, Algebra Commutativa e Aspetti Computazionali. The paper has been widely improved while the second author was guest at the University of Oslo. The financial support from both the University of Oslo and the Norwegian Research Council is gratefully aknowledged.