On endomorphisms of algebraic surfaces

2012

dimensional abelian varieties without non-trivial endomorphisms for every m> 1. This construction may be described as follows. Let Ka be an algebraic closure of a perfect field K with char(K) ̸ = 2. Let n = 2m + 1 or 2m + 2. Let us choose an n-element set R ∈ Ka that constitutes a Galois orbit in K and assume, in addition, that the Galois group of K(R) over K is “big ” say, coincides with full symmetric group Sn or the alternating group An. Let f(x) ∈ K[x] be the irreducible polynomial of degree n, whose set of roots coincides with R. Let us consider the hyperelliptic curve Cf: y 2 = f(x) over Ka and let J(Cf) be its jacobian which is the m-dimensional abelian variety. Then the ring End(J(Cf)) of all Ka-endomorphisms of J(Cf) coincides with Z if either n> 6 or char(K) ̸ = 3. The aim of this paper is to construct abelian threefolds without complex multiplication, using jacobians of non-hyperelliptic curves of genus 3. It is well-known that these curves are smooth plane quartics...

Compositio Mathematica, 2002

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n 5 b 2 À 2 disjoint smooth rational curves with self-intersection À2, where b 2 is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p g ¼ 0 and K 2 ¼ 8; 9 which admit an automorphism of order 2.

2002

In this note we shall determine all actions of groups of prime order p with p > 3 on Gorenstein del Pezzo (singular) surfaces Y of Picard number 1. We show that every order-p element in Aut(Y) (= Aut(Y'), Y' being the minimal resolution of Y) is lifted from a projective transformation of the projective plane. We also determine when

2001

We study the minimal complex surfaces of general type with pg = 0 and K 2 = 7 or 8 whose bicanonical map is not birational. We show that if S is such a surface, then the bicanonical map has degree 2 (see [MP1]) and there is a fibration f : S → P 1 such that: i) the general fibre F of f is a genus 3 hyperelliptic curve; ii) the involution induced by the bicanonical map of S restricts to the hyperelliptic involution of F. Furthermore, if K 2 S = 8, then f is an isotrivial fibration with 6 double fibres, and if K 2 S = 7, then f has 5 double fibres and it has precisely one fibre with reducible support, consisting of two components. 2000 Mathematics Classification: 14J29.

2021

This thesis will focus on the study of the relationship that exists between Del Pezzo surfaces, a kind of surface defined as "a smooth birationally trivial surface V on which the anticanonical sheaf is ample" and counting points in the projective plane P2(k) built over a finite field k. The result allowing us to to link the two concepts says that a Del Pezzo surface of degree d bigger that 1 is isomorphic to the blowup up of 9-d points in P2(k). Once we have settled the relationship between the two concepts the results will revolve around counting n-tuples of points in P2(k) both from a theoretical and computational point of view. In particular the case of 8-tuples, corresponding to Del Pezzo surfaces of degree 1, will be the one around which most of the work will revolve, culminating with the statement of a degree 8 monic polynomial expressing the number of $8$-tuples of points in general position as a function of the dimension of the base field. The work concludes by con...

2004

A minimal surface of general type with p g (S) = 0 satisfies 1 ≤ K 2 ≤ 9 and it is known that the image of the bicanonical map ϕ is a surface for K 2 S ≥ 2, whilst for K 2 S ≥ 5, the bicanonical map is always a morphism. In this paper it is shown that ϕ is birational if K 2 S = 9 and that the degree of ϕ is at most 2 if K 2 S = 7 or K 2 S = 8. By presenting two examples of surfaces S with K 2 S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K 2 S = 8 is, to our knowledge, a new example of a surface of general type with p g = 0. The degree of ϕ is also calculated for two other known surfaces of general type with p g = 0, K 2 S = 8. In both cases the bicanonical map turns out to be birational.

Bulletin of the American Mathematical Society, 1985

1999

A minimal surface of general type with pg(S) = 0 satisfies 1 ≤ K 2 ≤ 9 and it is known that the image of the bicanonical map φ is a surface for K2 S ≥ 2, whilst for K 2 S ≥ 5, the bicanonical map is always a morphism. Here we prove that if K2 = 7 or K2 = 8 then the degree of φ is at most 2 and that if K2 = 9, then φ is birational.

Journal of Algebra, 2001

We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X G and the quotient surface Y = X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori's extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonical divisors are used.