INTRODUCTION TO ALGEBRAIC GEOMETRY-page69

INTRODUCTION TO - 65 Example 7.6 We already know that P 1 k is isomorphic to a subvariety of P 2 k given by an equation of degree 2 This result can

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Unformatted text preview: 65 Example 7.6. We already know that P 1 k is isomorphic to a subvariety of P 2 k given by an equation of degree 2 . This result can be generalized as follows. Let N = ( n + m m )- 1 . Let us denote the projective coordinates in P N k by T i = T i ...i n , i + . . . + i n = | i | = m. Choose some order in the set of multi-indices i with | i | = m . Consider the morphism (the Veronese morphism of degree m) v n,m : P n k → P N k , defined by the collection of monomials ( . . . , T i , . . . ) of degree m . Since T i generate an irrelevant ideal, we can apply Proposition 17.4 , so this is indeed a morphism. For any k-algebra K the corresponding map v n,m ( K ) : P n k ( K ) → P N k ( K ) is defined by the formula ( a , . . . , a n ) → ( . . . , T i ( a ) , . . . ) . The image of v n,m ( K ) is contained in the set Ver m n ( K ) , where Ver m n is the projective subvariety of P N k given by the following system of homogeneous equations { T i T j- T k T t = 0 } i + j = k + t ....
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This note was uploaded on 01/08/2012 for the course MATH 299 taught by Professor Wei during the Spring '09 term at SUNY Stony Brook.

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