Threefolds Of General Type Research Articles

Toward classification of codimension 1 foliations on threefolds of general type

AbstractThe aim of this paper is to classify codimension 1 foliations with canonical singularities and on threefolds of general type. I prove a classification result for foliations satisfying these conditions and having nontrivial algebraic part. We also describe purely transcendental foliations with the canonical class being not big on manifolds of general type in any dimension, assuming that is nonsingular in codimension 2.

Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic

Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic

Threefolds of general type on the Noether line

Threefolds of general type on the Noether line

Examples of surfaces with canonical map of maximal degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois etale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois etale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

Examples of surfaces with canonical map of maximal degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois etale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois etale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

On the canonical degrees of Gorenstein threefolds of general type

Let X be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose that the canonical map is generically finite onto its image. C. Hacon showed that the canonical degree is universally bounded by 576. We improved Hacon’s universal bound to 360. Moreover, we gave all the possible canonical degrees of X if X is an abelian cover over \(\mathbb {P}^3\) and constructed all the examples with these canonical degrees.

The third and fourth pluricanonical maps of threefolds of general type

AbstractFor a nonsingular projective threefold of general typeXover the field of complex numbers, we show that the fourth pluricanonical map ϕ4is not birational onto its image if and only ifXis birationally fibred by (1,2)-surfaces, provided that vol(X) ≥ 303. We also have similar characterization of birationality of ϕ3.

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 pg=0 and irregularities q1=q2=0 have bigenus P2≤6. In the present paper nonsingular threefolds with pg=q1=q2=0 and P2=7 or P2=8 are constructed.

On threefolds of general type with χ = 1

For a canonical threefold of general type, we know that the pluri–canonical map \({\phi_{n}}\) is stably birational for a sufficiently large n. This paper aims to find the lower bound of n for such kind of threefolds with χ = 1. To prove our main result, we will estimate the lower bound of plurigenus.

A Threefold of General Type with q1=q2=pg=P2=0

One of the problems in classifying nonsingular threefolds of general type with pg=0 lies in finding the range of the bigenus P2 (surfaces of general type with pg=0 have 2≤P2≤10). Another problem involves finding the minimum integer m such that the m-canonical map Φ|mK| is birational for any threefold (m=5 in the case of surfaces). An example of a nonsingular threefold X of general type with q1=q2=0, pg=P2=0,P3=1 is presented. In addition, the m-canonical map of X is birational if and only if m≥14. The threefold is obtained as a nonsingular model of a degree ten hypersurface in P4C with the affine equation t2=f10(x,y,z).