A Threefold of General Type with q 1=q 2=p g =P 2=0

One of the problems in classifying nonsingular threefolds of general type with p _g =0 lies in finding the range of the bigenus P ₂ (surfaces of general type with p _g =0 have 2P ₂10). Another problem involves finding the minimum integer m such that the m-canonical map _|mK| is birational for any threefold (m=5 in the case of surfaces). An example of a nonsingular threefold X of general type with q ₁=q ₂=0, p _g =P ₂=0,P ₃=1 is presented. In addition, the m-canonical map of X is birational if and only if m14. The threefold is obtained as a nonsingular model of a degree ten hypersurface in P ⁴ _C with the affine equation t ²=f ₁₀(x,y,z).     

On the geography of threefolds of general type

Let X be a complex nonsingular projective 3-fold of general type. We show that there are positive constants c, $c^{\prime }$ and $m_1$ such that $\chi ({\omega }_X)??c\mathrm{Vol}(X)$ and $P_m(X)?c^{\prime }{m}^3\mathrm{Vol}(X)$ for all $m?m_1$.     

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 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 .

     

Characterization of a class of surfaces with p g = 0 and K 2 = 5 by their bicanonical maps

Let S be a minimal surface of general type with ${p_g(S) = 0, K_S^2 = 5}$ . We prove that S is a Burniat surface if its bicanonical map is of degree 4 and has smooth image.     

A new family of surfaces of general type with K^2=7 and p_g=0

We construct a new family of smooth minimal surfaces of general type with $K^2=7$ and $p_g=0$ . We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.     

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...     

On surfaces with p g =2, q=1 and K 2=5

We consider minimal surfaces of general type with p _g =2, q=1 and K ²=5. We provide a stratification of the corresponding moduli space \(\mathcal{M}\) and we give some bounds for the number and the dimensions of its irreducible components.     

Projective threefolds on which acts with 2-dimensional general orbits

The birational geometry of projective threefolds on which acts with 2-dimensional general orbits is studied from the viewpoint of the minimal model theory of projective threefolds. These threefolds are closely related to the minimal rational threefolds classified by Enriques, Fano and Umemura. The main results are (i) the -birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.