On semi‐isogenous mixed surfaces

Let C be a smooth projective curve and G be a finite subgroup of whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of . Let be the index two subgroup . If G⁰ acts freely, then is smooth and we call it semi‐isogenous mixed surface. In this paper we give an algorithm to determine semi‐isogenous mixed surfaces with given geometric genus, irregularity and self‐intersection of the canonical class. As an application we classify irregular semi‐isogenous mixed surfaces with and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with . We provide an example of a minimal surface of general type with and .     

The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil

Received: 15 October 1999; in final form: 13 June 2000 / Published online: 29 April 2002     

Infinitesimal Torelli theorem for surfaces of general type with certain invariants

We prove the infinitesimal Torelli theorem for general minimal complex surfaces X's with the first Chern number 3, geometric genus 1, and irregularity 0 which have non-trivial 3-torsion divisors. We also show that the coarse moduli space for surfaces with the invariants as above is a 14-dimensional unirational variety.     

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.     

On the moduli of surfaces admitting genus 2 fibrations

We investigate the structure of the components of the moduli space of surfaces of general type, which parametrize surfaces admitting nonsmooth genus 2 fibrations of nonalbanese type, over curves of genusg _b≥2.     

On a map between two K3 surfaces associated to a net of quadrics

We study the geometry of the birational map between an intersection of a net of quadrics in that contains a line and the double sextic branched along the discriminant of the net. We show that the branch locus of a smooth double sextic S ₆ is discriminant of a net of quadrics in such that S ₆ is isomorphic to the intersection of this net iff a certain configuration of rational curves on S₆ is weakly even. Received: 14 September 2005 Suported by the DFG Schwerpunktprogramm ‘Global methods in complex geometry’. The first named author is partially supported by the KBN Grant No. 1 P03A 008 28. The second named author is partially supported by the KBN Grant No. 2 P03A 016 25.     

A fake projective plane with an order 7 automorphism

A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface.In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface.     

K3 surfaces with an order 50 automorphism

In any characteristic p different from 2 and 5, Kondō gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic $p\ne 2,5$, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kondō.     

The alternating groups and K3 surfaces

In this note, we consider all possible extensions G of a non-trivial perfect group H acting faithfully on a K3 surface X. The pair (X,G) is proved to be uniquely determined by G if the transcendental value of G is maximum. In particular, we have , if H is the alternating group A₅ and normal in G.     

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.     

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.     

Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds

We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the rational-connectedness conjecture in Kollar et al. (J. Algebra Geom. 1 (1992) 429) which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group.     

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.     

Seshadri constants in a family of surfaces

The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. Received: 16 April 2001 / Published online: 26 April 2002