Algebraic Curves: An Introduction to Algebraic GeometryBenjamin, 1969 - 226 pages |
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Page 64
... called a or simple point of F if either derivative Fx ( P ) # 0 Fy ( P ) 0 . In this case the line F ( P ) ( X - a ) + Fy ( P ) ( Y - b ) = 0 is called the tangent line to F at P. A point which isn't simple is called multiple ( or ...
... called a or simple point of F if either derivative Fx ( P ) # 0 Fy ( P ) 0 . In this case the line F ( P ) ( X - a ) + Fy ( P ) ( Y - b ) = 0 is called the tangent line to F at P. A point which isn't simple is called multiple ( or ...
Page 66
... called tangent to the curve at ( 0,0 ) . Let F be any curve , P ( 0,0 ) . Write F = F + F m Fm + 1 of degree i , = + . + F where F F 0 . m of F at P = ( 0,0 ) , Pε F if and only if n ' We define and write m ( F ) > 0 . m ( F ) > 0 . i ...
... called tangent to the curve at ( 0,0 ) . Let F be any curve , P ( 0,0 ) . Write F = F + F m Fm + 1 of degree i , = + . + F where F F 0 . m of F at P = ( 0,0 ) , Pε F if and only if n ' We define and write m ( F ) > 0 . m ( F ) > 0 . i ...
Page 89
... called an algebraic set in p or a projective algebraic set . For any set Xc p " , p1 , we let I ( X ) = { F ε k [ X1 , ··· , Xn + 1 ] ] every P ɛ X is a zero of F } . I ( X ) is called the ideal of X. An ideal I c k [ X1 , ... , xn + 1 ...
... called an algebraic set in p or a projective algebraic set . For any set Xc p " , p1 , we let I ( X ) = { F ε k [ X1 , ··· , Xn + 1 ] ] every P ɛ X is a zero of F } . I ( X ) is called the ideal of X. An ideal I c k [ X1 , ... , xn + 1 ...
Table des matières
Chapter One Affine Algebraic Sets 1 Algebraic Preliminaries | 1 |
Affine Space and Algebraic Sets | 7 |
The Ideal of a Set of Points | 10 |
Droits d'auteur | |
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Expressions et termes fréquents
affine variety algebraic set algebraic subset algebraically closed assume Bezout's Theorem birational birationally equivalent called change of coordinates Chapter closed subvariety comaximal COROLLARY curve F curve of degree defined deg(D deg(F denote div(G div(z divisor element F and G f ɛ finite number follows form of degree function field Hint homogeneous hyperplane hypersurface induced integer intersection number isomorphism k[X₁ LEMMA Let F linear local ring maximal ideal module-finite morphism mp(F Noether's non-singular non-zero Nullstellensatz Op(C Op(V open subvariety ordinary multiple points plane curve point on F polynomial map Problem projective change projective curve projective plane curve projective variety Proof Prop PROPOSITION quadratic transformation quotient field R-module rational function residue ring homomorphism simple point subring subvariety Suppose tangent line uniformizing parameter unique V₁ vector space Xn+1 zero Ρε