We prove the boundedness of complements modulo two conjectures: Borisov-Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.We give a sketch of the proof of our main results in Section 4. One can see that our proof essentially uses reduction to lower-dimensional global pairs. However it is expected that an improvement of our method can use reduction to local questions in the same dimension. In fact we hope that the hypothesis in our main theorem 1.4 should be the existence of local complements and Conjecture 1.1 for ε-lt Fano varieties (without a boundary), where ε ≥ ε o > 0, ε o is a constant depending only on the dimension (cf. Addendum 1.6 and Corollary 1.7). If dim X = 2, we can take ε o = 1/7. Note also that our main theorem 1.4 is weaker than one can expect. We think that the pair (X, B) can be taken arbitrary log-semi-Fano (possibly not klt and not FT) and possible boundary multiplicities can be taken arbitrary real numbers in [0, 1] (not only in Φ(R)). The only hypothesis we have to assume is the existence of an Rcomplement B + ≥ B (cf. [Sho00]). However the general case actually needs a finite set of natural numbers for complements, and there are no such universal number for all complements (cf. [Sho93, Example 5.2.1]). Preliminaries2.1. Notation. All varieties are assumed to be algebraic and defined over an algebraically closed field k of characteristic zero. Actually, the main results hold for any k of characteristic zero not necessarily algebraically closed since they are related to singularities of general members of linear systems (see [Sho93, 5.1]). We use standard terminology and notation of the Log Minimal Model Program (LMMP) [KMM87], [Kol92], and [Sho93]. For the definition of complements and their properties we refer to [Sho93], [Sho00], [Pro01] and [PS01]. Recall that a log pair (or a log variety) is a pair (X, D) consisting of a normal variety X and a boundary D, i.e., an R-divisor D = d i D i with multiplicities 0 ≤ d i ≤ 1. As usual K X denotes the canonical (Weil) divisor of a variety X. Sometimes we will write K instead of K X if no confusion is likely. Everywhere below a(E, X, D) denotes the discrepancy of E with respect to K X + D. Recall the standard notation:In this paper we use the following strong version of ε-log terminal and ε-log canonical property. Definition 2.2. A log pair (X, B) is said to be ε-log terminal (ε-log canonical) if totaldiscr(X, B) > −1 + ε (resp., totaldiscr(X, B) ≥ −1 + ε). Licensed to New York Univ, Courant Inst. Prepared on Sat Jul 4 02:39:10 EDT 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf TOWARDS THE SECOND MAIN THEOREM ON COMPLEMENTS 155 Usually we work with R-divisors. An R-divisor is an R-linear combination of prime Weil divisors. AnTwo R-divisors D and D are said to be Q-(resp., R-)linearly equivalent if D − D is a Q-(resp., R-)linear combination of principal divi...

We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a Q-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold X of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of X is very ample except for some explicit cases.We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use it to identify some of them.