dc.contributor.advisor Cheltsov, Ivan en dc.contributor.advisor Gordon, Iain en dc.contributor.author Karzhemanov, Ilya en dc.date.accessioned 2019-01-15T11:20:05Z dc.date.available 2019-01-15T11:20:05Z dc.date.issued 2010-11-24 dc.identifier.uri http://hdl.handle.net/1842/33321 dc.description.abstract The thesis consists of four chapters. First chapter is introductory. In Chapter 2, we recall some basic facts from the singularity theory of algebraic varieties (see Section en 2.2) and the theory of minimal models (see Section 2.3), which will be used throughout the rest of the thesis. We also make some conventions on the notions and notation used in the thesis (see Section 2.1). Each Chapter 3 and 4 starts with some preliminary results (see Sections 3.1 and 4.1, respectively). Each Chapter 3 and 4 ends with some corollaries and conclusive remarks (see Sections 3.7 and 4.4, respectively). In Chapter 3, we prove Theorem 1.2.7, providing the complete description of Halphen pencils on a smooth projective quartic threefold X in P4. Let M be such a pencil. Firstly, we show that M ⊂ | − nKX | for some n ∈ N, and the pair (X,1n M) is canonical but not terminal. Further, if the set of not terminal centers CS(X, 1 ) (see Remark 2.2.8) does not contain points, we show that n = 1 (see Section 3.2). Finally, if there is a point P ∈ CS(X, n M), in Section 3.1 we show first that a general M ∈ M has multiplicity 2n at P (cf. Example 1.2.3). After that, analyzing the shape of the Hessian of the equation of X at the point P , we prove that n = 2 and M coincides with the exceptional Halphen pencil from Example 1.2.6 (see Sections 3.3–3.6). In Chapter 4, we prove Theorem 1.2.11, which shows, in particular, that a general smooth K3 surfaces of type R is an anticanonical section of the Fano threefold X with canonical Gorenstein singularities and genus 36. In Section 4.2, we prove that X is unique up to an isomorphism and has a unique singular point, providing the geometric quotient construction of the moduli space F in Section 4.3 (cf. Remark 1.2.12). Finally, in Section 4.3 we prove that the forgetful map F −→ KR is generically surjective. dc.language.iso en dc.publisher The University of Edinburgh en dc.subject singularity theory of algebraic varieties en dc.title Fano threefolds and algebraic families of surfaces of Kodaira dimension zero en dc.type Thesis or Dissertation en dc.type.qualificationlevel Doctoral en dc.type.qualificationname PhD Doctor of Philosophy en
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