Complements on surfaces

Abstract

The main result of the paper is a boundedness theorem for ncomplements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated 1 2 .

Complements on log surfaces

“…In many cases this theorem is true without condition (4). When proving this theorem we follow the paper [16]. The cases using condition (4) are considered in details.…”

“…The theory of complements on algebraic varieties has been created by V. V. Shokurov in the papers [15], [16]. It is a powerful tool for studying algebraic varieties, extremal contractions and singularities.…”

“…The cases δ(S, D) = 1 and δ(S, D) = 2 were classified in the papers [9], [16]. To study the remaining case δ(S, D) = 0 the theory of complements on surfaces must be applied in more wide set of coefficients.…”

On the classification of Mori contractions: the case of an elliptic curve

“…A contraction f : X → Z ∋ o such as in 1.1 is said to be exceptional if for any complement K X + D near f −1 (o) there is at most one divisor E of the function field K(X) with discrepancy a(E, D) = −1. The following is a particular case of the theorem proved in [Sh1] and [P2] (see also [PSh]). Theorem 1.2.…”

“…He proved that δ ≤ 2, classified log surfaces with δ = 2 and showed that in the case δ = 1 the (unique) divisor E with a(E, ∆) ≤ −6/7 is represented by a curve of arithmetical genus ≤ 1 (see [Sh1], [P3]). The aim of this short note is to exclude the case of Mori contractions with δ = 1 and elliptic curve E: Theorem 1.5.…”