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Week
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Topics
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Study Metarials
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1
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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
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2
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Algorithmic computations to obtain the simple representatives of quotients of polynomial rings
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3
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The coordinate ring of an affine variety, the notion of isomorphism between the varieties and its relations with the coordinate rings
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4
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The function field of a variety, rational functions between the varieties and the notion of birational equivalence
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5
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Notion of the point at infinity, projective plane, projective space and the homogenous coordinates, covering of the projective space by affine spaces
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6
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The definition and examples of projective variety, the relation between a projective variety and the homogenous ideals, projective Hilbert`s Nullstellensatz
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7
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Projective closure of an affine variety, morphisms between the projective varieties, Veronese and Segre functions
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8
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Projective elimination theory
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9
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Automorphisms of a projective space and projectively equivalent varieties, projective equivalence of hyperplanes
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10
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Geometry of the quadric hypersurfaces
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11
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Variety of a monomial ideal, its dimension and its complement
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12
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The affine Hilbert function and the affine Hilbert polynomial of an ideal, the dimension of an affine variety and of a projective variety
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13
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Basic properties of the dimension, the relation between the dimension and the algebraic independence
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14
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The relation between the dimension and nonsingularity
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