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).     

A bound for the orders of the torsion groups of surfaces with c 1 2 = 2 - 1

We shall give a bound for the orders of the torsion groups of minimal algebraic surfaces of general type whose first Chern numbers are twice the Euler characteristics of the structure sheaves minus 1, where the torsion group of a surface is the torsion part of the Picard group. Namely, we shall show that the order is at most 3 if the Euler characteristic is 2, that the order is at most 2 if the Euler characteristic is greater than or equal to 3, and that the order is 1 if the Euler characteristic is greater than or equal to 7.     

Higher dimensional Calabi–Yau manifolds of Kummer type

Based on Cynk–Hulek method from [7] we construct complex Calabi–Yau varieties of arbitrary dimensions using elliptic curves with an automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize the result of [11] to obtain arbitrarily dimensional Calabi–Yau manifolds which are Zariski in any characteristic $p\not\equiv \text{◂⋅▸}1(\mathrm{mod}12)$.     

Complex surfaces of general type with K2 = 3,4 and pg = q = 0

We construct complex surfaces of general type with $p_g=0$ and $K^2=3,4$ as double covers of Enriques surfaces (called Keum–Naie surfaces) with a different way to the original constructions of Keum and Naie. As a result, we show that there is a $(?4)$-curve on the example with $K^2=3$, which might imply a special relation between Keum–Naie surfaces with $K^2=3$ and $K^2=4$.     

Surfaces with pg = 2, K2 = 3 and a pencil of curves of genus 2

We classify minimal smooth surfaces of general type with K ² = 3, p _g = 2 which admit a fibration of curves of genus 2.We prove that they form an irreducible set of dimension 22 in their moduli space.      

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(S) = 0 satisfies 1 K² 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

Surfaces with p g = q = 1, K 2 = 6 and Non-birational Bicanonical Maps

Let S be a smooth minimal projective surface of general type with p_g(S) = q(S) = 1,K_S~2= 6. We prove that the degree of the bicanonical map of S is 1 or 2. So if S has non-birational bicanonical map, then it is a double cover over either a rational surface or a K3 surface.     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7, II

Minimal complex surfaces of general type with p_g = 0 and K²= 7 or 8 whose bicanonical map is not birational are studied.It is shown that if S is such a surface, then the bicanonicalmap has degree 2 (see Bulletin of the London Mathematical Society33 (2001) 1–10) and there is a fibration f: S P¹ suchthat (i) the general fibre F of f is a genus 3 hyperellipticcurve; (ii) the involution induced by the bicanonical map ofS restricts to the hyperelliptic involution of F. Furthermore, if , then f isan isotrivial fibration with six double fibres, and if , then f has five double fibres andit has precisely one fibre with reducible support, consistingof two components. 2000 Mathematics Subject Classification 14J29.     

Godeaux,Campedelli, and surfaces of general type with and

We construct simply connected surfaces of general type with invariants and . We use ‐Gorenstein deformations in conjunction with explicit constructions that express the canonical rings by generators and relations. The canonical rings of the surfaces are described as projections. The whole construction is simplified by the use of key varieties based on Steiner 3‐folds. As a consequence of the construction we find two families, each family in a different connected component of the moduli stack , and each linking a Campedelli surface with a Godeaux surface.     

A family of surfaces with and Albanese map of degree 3

We study a family of surfaces of general type with and , originally constructed by Cancian and Frapporti by using the Computer Algebra System MAGMA . We provide an alternative, computer‐free construction of these surfaces, that allows us to describe their Albanese map and their moduli space.     

Certain algebraic surfaces with Eisenbud‐Harris general fibration of genus 4

We classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.     

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$-actions and the locus of Gorenstein stable surfaces

In this note, the geography of minimal surfaces of general type admitting ${\mathbb{Z}}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ contains surfaces admitting a ${\mathbb{Z}}_2^2$-action for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$ or $K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification ${\overline{\mathfrak{M}}}_{K^2,\chi }$ of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$.     

Exceptional curves on smooth rational surfaces with not nef and of self-intersection zero

A -curve is a smooth rational curve of self-intersection , where is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has -curves. In this paper we prove that for such a surface , the set of -curves on is finite but non-empty, and that may have no -curves. Related facts are also considered.