De-Qi Zhang

Gorenstein log del Pezzo Surfaces of Rank One

Let S be a projective normal surface defined over the field C of complex numbers. We say that S is a log de1 Pezzo surface provided S has only quotient singularities and the anticanonical divisor -X, is ample. Note that if S has only quotient singularities, a multiple XA of every Weil divisor A on S becomes a Cartier divisor for some integer N determined by S and hence the intersection theory of Weil divisors is available (cf. Miyanishi and Tsunoda [S]). The rank of S is the Picard rank p(S) = dimo Pit(S) @ Q. Moreover, S is Gorenstein, i.e., K, is a Cartier divisor, if and only if S has only rational double points provided S is a log de1 Pezzo surface. Let p: Y-+ S be the minimal resolution of singularities and let D be the exceptional locus, which we identify with an effective reduced divisor with support D. According to the terminology of [8], a pair (V, D) is a log del Pezzo surface of rank one with contractible boundary iff S is a log dei Pezzo surface of rank one in the above sense. En the present article, we are mainly interested in the case where S is a Gorenstein log de1 Pezzo surface. As for the construction of such a surface, Demazure [j] has shown the existence of a birational morphism G: V+ P*, unless S is P’ x P’ or a quadric cone, which is a composite of at most 8 blowing-ups (cf. Hidaka and Watanabe [7]). They have also shown that 1 --I&I admits a smooth member. On the other hand, Brenton [3] and Bindschadler and Brenton [2] observed topological properties of S and Gorenstein compactification of C*. Though the subject treated in the present article is related to theirs, our approach is more algebro-geometric and different from their topological ones. We shall prove that:

Building blocks of étale endomorphisms of complex projective manifolds

Etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak Calabi-Yau varieties.

Classification of extremal elliptic {

We present a complete list of extremal elliptic K3 surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial (Proposition 4.1 and Theorems 4.3).

K3

A Coble surface is a smooth rational projective surface such that its anti-canonical linear system is empty while the anti-bicanonical linear system is nonempty. In this paper we shall classify Coble surfaces and consider the finiteness problem of the number of negative rational curves on it modulo automorphisms.

} surfaces and fundamental groups of open {

We shall give a proof for Vorontsov’s Theorem and apply this to classify log Enriques surfaces with large prime canonical index.

K3

The aim of this note is to characterize a K3 surface of Klein-Mukai type in terms of its symmetry.

} surfaces

We show that the M-canonical map of an n-dimensional complex projective manifold X of Kodaira dimension two is birational to an Iitaka fibration for a computable positive integer M. M depends on the index b of a general fibre F of the Iitaka fibration and on the Betti number of the canonical covering of F, In particular, M is a universal constant if the dimension n is smaller than or equal to 4.

Coble rational surfaces

Abstract 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.

On Vorontsov's Theorem on K3 surfaces with non-symplectic group actions

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.

THE SIMPLE GROUP OF ORDER 168 AND K3 SURFACES

This theorem is a generalization of a result in [5], where we proved the finiteness of 7Tj((S°) under a stronger condition that S is a log del Pezzo surface. (See also [4] for a differential geometric proof in the log del Pezzo surface case.) One may try to reduce the above theorem to the case where the anti-canonical divisor is ample. Namely, let/S-^Tbe the contraction of all curves numerically perpendicular to — Ks. However, T may have worse than quotient singularities and n^T) may not be finite. Recall that a singularity has a finite local n1 if and only if it is a quotient singularity (cf. [1, Satz 2-8]). In [10], we shall consider log canonical singularity case. The nef and bigness of —^5 implies that the anti-canonical Kodaira dimension K(S, —KS) = 2. But one cannot relax the condition in the theorem to K(S, —KS) = 2 because there are (smooth) non-rational ruled surface with anticanonical dimension 2 (cf. [8], [9]). Neither can one remove the bigness condition in the theorem. Indeed, just consider the example 8 = [E x P)/r as in [5] where r is an involution acting diagonally and non-trivially on both the elliptic curve E and P. In this case, S is a rational surface with only rational double point singularities and — Ks is nef with (-Ks) 2 = 0 but n^S) is infinite.