Topics in modern algebraic geometry

In section one we consider an algebraic variety X over an algebraically closed field k  and then study the non-standard enlarged variety *X.  We study the shadow of internal subvarieties of *X  (theorem 1.3.6).  We prove the Nullstellensatz for infinite dimensional varieties (theorem 1.5.3).  Then we study the enlargement of commutative rings. Section two we give a survey of the fundamental paper of Shokurov [Sh4] on the existence of log flips in dimension 3 (no result of mine in section two).  In section three we outline Shokurov's program (see 3.1.7.1) to attack the log termination conjecture (3.1.7), the ACC conjecture on mlds (3.1.6) and the Alexeev-Borisov conjecture on the boundedness of δ-1c Fano varieties (3.1.6) in higher dimensions.  The core of this program is the boundedness of ε-lc complements conjecture due to Shokurov (conjecture 3.1.2). We prove the later conjecture in dimension two (theorem 3.7.1 and theorem 3.10.1).  We give a totally new proof to the Alexeev-Borisov conjecture in dimension two (corollary 3.7.9).  We also prove that the boundedness of lc complements due to Shokurov (theorem 3.1.15) can be proved only using the theory of complements.  Our most important result is the method used to prove the boundedness of ε-lc complements conjecture (3.7.1 and 3.10.1). In section four we outline separate plans proposed by the author of this thesis and Shokurov toward the boundedness of ε-lc complements conjecture in dimension three.