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Prerequisites
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Language of Instruction
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Turkish
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Responsible
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Assist. Prof. Dr. Celalettin KAYA
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Instructors
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Assistants
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The related lecturers of the department
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Resources
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[1] Ideals, Varieties, and Algorithms (4th Edition), David Cox, John Little, Donal O. Shea, Springer-Verlag, New York, 2015.
[2] Undergraduate Algebraic Geometry, Miles Reid, Cambridge University Press, London, 1992.
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Supplementary Book
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1. An Invitation To Algebraic Geometry, Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves, Springer-Verlag, New York, 2010.
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Goals
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To investigate the geometry of varieties by the help of functions defined on them, to study the dimension and properties of a variety, to teach the basic notions of projective algebraic geometry.
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Content
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The notion of a regular function between varieties and the information it gives about the geometric properties of the variety; Basic definitions and results on the quotients of polynomial rings; Algorithmic computations to obtain the simple representatives of quotients of polynomial rings; The coordinate ring of an affine variety, the notion of isomorphism between the varieties and its relations with the coordinate rings; The function field of a variety, rational functions between the varieties and the notion of birational equivalence; Notion of the point at infinity, projective plane, projective space and the homogenous coordinates, covering of the projective space by affine spaces; The definition and examples of projective variety, the relation between a projective variety and the homogenous ideals, projective Hilbert`s Nullstellensatz; Projective closure of an affine variety, morphisms between the projective varieties, Veronese and Segre functions; Projective elimination theory; Automorphisms of a projective space and projectively equivalent varieties, projective equivalence of hyperplanes; Geometry of the quadric hypersurfaces; Variety of a monomial ideal, its dimension and its complement; The affine Hilbert function and the affine Hilbert polynomial of an ideal, the dimension of an affine variety and of a projective variety; Basic properties of the dimension, the relation between the dimension and the algebraic independence; The relation between the dimension and nonsingularity.
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