Kawamata–Viehweg vanishing on rational surfaces in positive characteristic

We prove that the Kawamata–Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W ₂(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata–Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

Birationally rigid del Pezzo fibrations

In this paper we study del Pezzo fibrations z:X¹ of degree 1 and 2 such that X is smooth, rk Pic(X)=2 and These are examples of smooth birationally rigid 3-fold Mori fibre spaces. We describe all birational transformations of the 3-fold X into elliptic fibrations, fibrations of surfaces of Kodaira dimension zero, and canonical Fano 3-folds.     

Regularization of Birational Automorphisms

An effective method for regularizing birational automorphisms of multidimensional algebraic varieties is suggested and applied explicitly to some three-dimensional Fano varieties and del Pezzo surfaces over an algebraically nonclosed field.     

Complements on surfaces

The main result is a boundedness theorem forn-complements on algebraic surfaces. In addition, this theorem is used in a classification of log Del Pezzo surfaces and birational contractions for threefolds. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 62, Algebraic Geometry-10, 1999.     

Log canonical thresholds of del Pezzo surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Kollár–Shokurov’s connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic 0. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degree. As an application we conclude that non-singular del Pezzo surfaces in finite characteristic of degree lower or equal than 4 are K-semistable.     

DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

Abstract

The purpose of this paper is to extend the result in Park (Park, J. (2001). Birational maps of del Pezzo fibrations. J. Reine Angew. Math. 538:213–221) in the case of del Pezzo fibrations of degree 1. To this end we investigate the anticanonical linear systems of del Pezzo surfaces of degree 1. We then classify all possible effective anticanonical divisors on Gorenstein del Pezzo surfaces of degree 1 with canonical singularities.     

Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series

Abstract

The Hilbert series of a polarized algebraic variety (X, D) is a powerful invariant that, while it captures some features of the geometry of (X, D) precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case.     

On Fibrations into del Pezzo Surfaces

The problem of birational rigidity for three-dimensional algebraic varieties fibered over rational curves into del Pezzo surfaces of degree 1 is discussed. A criterion for the rigidity of such fibrations in the Mori category is suggested and the inverse implication is proved (Theorem 3.3). Surgeries on fibers in fibrations of this type, which turn out to be closely related to the rigidity problem, are considered. In particular, an important result on the uniqueness of a smooth model in a class of maps over a base is stated (Corollary 4.5).     

On Nonrational Fibers of del Pezzo Fibrations over Curves

We consider threefold del Pezzo fibrations over a curve germ whose central fiber is nonrational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a one-to-one correspondence between such fibrations and certain nonsingular del Pezzo fibrations equipped with a cyclic group action.

     

Varieties birationally isomorphic to affine <Emphasis Type="Italic">G</Emphasis>-varieties

Let a linear algebraic group G act on an algebraic variety X. Classification of all these actions, in particular birational classification, is of great interest. A complete classification related to Galois cohomologies of the group G was established. Another important question is reducibility, in some sense, of this action to an action of G on an affine variety. It has been shown that if the stabilizer of a typical point under the action of a reductive group G on a variety X is reductive, then X is birationally isomorphic to an affine variety [`(X)] \bar X with stable action of G. In this paper, I show that if a typical orbit of the action of G is quasiaffine, then the variety X is birationally isomorphic to an affine variety [`(X)] \bar X .     

Seshadri constants on rational surfaces with anticanonical pencils

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Second, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.     

Del Pezzo surfaces of degree 6 over an arbitrary field

We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field F. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen K-theory of the surface. As a consequence, we obtain an index reduction formula for the function field of the surface.     

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.     

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index ℓ. This upper bound turns out to be a quadratic polynomial in the variable ℓ. Received: 18 June 2008     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

Birational aspects of the geometry of Varieties of Sums of Powers

Varieties of Sums of Powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms. The study of these varieties dates back to Sylvester and Hilbert, but only few of them, for special degrees and number of variables, are concretely identified. In this paper we aim to understand a general birational behavior of VSP. To do this we birationally embed these varieties into Grassmannians and prove the rational connectedness of many VSP in arbitrary degrees and number of variables.