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INTRODUCTION TO ALGEBRAIC GEOMETRY-page69

INTRODUCTION TO ALGEBRAIC GEOMETRY-page69 - 65 Example 7.6...

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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 0 ...i n , i 0 + . . . + 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 0 , . . . , 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 . It is called the m -fold Veronese variety of dimension n . We claim that the image of v n,m ( K ) is equal to Ver m n ( K ) for all K, so that v n,m defines an isomorphism of
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