Extremal metrics on del Pezzo threefolds

We prove the existence of Kähler-Einstein metrics on a nonsingular section of the Grassmannian Gr(2, 5) ? ?⁹ by a linear subspace of codimension 3 and on the Fermat hypersurface of degree 6 in ?(1, 1, 1, 2, 3). We also show that a global log canonical threshold of the Mukai-Umemura variety is equal to 1/2.     

Del Pezzo surfaces of degree 1 and Jacobians

We construct absolutely simple jacobians of nonhyperelliptic genus 4 curves, using Del Pezzo surfaces of degree 1.     

On the existence of almost Fano threefolds with del Pezzo fibrations

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.     

Polygonal cycles in higher Chow groups of Jacobians

The aim of this paper is to construct non-trivial cycles in the first higher Chow group of the Jacobian of a curve having special torsion points. The basic tool is to compute the analogue of the Griffiths infinitesimal invariant of the natural normal function defined by the cycle as the curve moves in the corresponding moduli space. We prove also a Torelli-like theorem. The case of genus 2 is considered in the last section. To the memory of Fabio BardelliMathematics Subject Classification (2000) 14C25, 14C34     

Normal del Pezzo surfaces containing a nonrational singularity

Working over perfect ground fields of arbitrary characteristic, I classify minimal normal del Pezzo surfaces containing a nonrational singularity. As an application, I determine the structure of 2-dimensional anticanonical models for proper normal algebraic surfaces. The anticanonical ring may be non-finitely generated. However, the anticanonical model is either a proper surface, or a proper surface minus a point. Received: 5 June 2000 / Revised version: 10 November 2000     

Log del Pezzo surfaces with not small fractional indices

For a log del Pezzo surface S, the fractional index is the maximum of r with which can be written as r times some Cartier divisor. We classify all the log del Pezzo surfaces S with , after the technique of Nakayama.     

Homogeneous spaces and degree 4 del Pezzo surfaces

It is known that, given a genus 2 curve , where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space for complete 2-descent on the Jacobian of , there is a V _δ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that . We shall prove that every degree 4 del Pezzo surface V, defined over K, arises in this way; furthermore, we shall show explicitly how, given V, to find and δ such that V =  V _δ , up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann’s example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over , whose Jacobians have nontrivial members of the Shafarevich-Tate group. This example will differ from previous examples in the literature by having only two -rational Weierstrass points. The author thanks EPSRC for support: grant number EP/F060661/1.     

Exceptional sets on del Pezzo surfaces with one log-terminal singularity

New full exceptional sets of coherent sheaves on a certain family of log-terminal del Pezzo surfaces, which is treated as a smooth stack, are constructed. These surfaces are not toroidal and can be represented as hypersurfaces in weighted projective 3-space.     

Drinfeld modules and torsion in the Chow groups of certain threefolds

Let E B be an elliptic surface defined over the algebraic closureof a finite field of characteristic greater than 5. Let W bea resolution of singularities of E x _B E. We show that the l-adicAbel–Jacobi map from the l-power-torsion in the secondChow group of W to H³(W, _l(2)) _l/_l is an isomorphism for almostall primes l. A main tool in the proof is the assertion thatcertain CM-cycles in fibres of W B are torsion, which is provenusing results from the theory of Drinfeld modular curves.     

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of \mathbbG_\texta² \mathbb{G}_{\text{a}}^2 , to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \mathbbG_\texta {\mathbb{G}_{\text{a}}} ⋊ \mathbbG_\textm {\mathbb{G}_{\text{m}}} . Bibliography: 32 titles.     

Weak approximation on del Pezzo surfaces of degree 1

We study del Pezzo surfaces of degree 1 of the form

w²=z³+Ax⁶+By⁶     

Degeneration of torsors over families of del Pezzo surfaces

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has $\mathbf{ADE}$-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

Quartic del Pezzo surfaces over function fields of curves

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.     

Log del Pezzo surfaces of rank 2 and Cartier index 3 with a unique singularity

Log del Pezzo surfaces play the role of the opposite of surfaces of general type. We will completely classify all the log del Pezzo surfaces of rank 2 and Cartier index 3 with a unique singularity.