By Joe Harris

ISBN-10: 0387977163

ISBN-13: 9780387977164

This booklet relies on one-semester classes given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. it's meant to be, because the name indicates, a primary advent to the topic. in spite of this, a couple of phrases are so as concerning the reasons of the booklet. Algebraic geometry has constructed greatly during the last century. in the course of the nineteenth century, the topic was once practiced on a comparatively concrete, down-to-earth point; the most gadgets of research have been projective types, and the strategies for the main half have been grounded in geometric structures. This process flourished throughout the center of the century and reached its fruits within the paintings of the Italian tuition round the finish of the nineteenth and the start of the twentieth centuries. eventually, the topic was once driven past the bounds of its foundations: through the top of its interval the Italian institution had improved to the purpose the place the language and methods of the topic may now not serve to precise or perform the guidelines of its most sensible practitioners.

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**Extra resources for Algebraic Geometry: A First Course**

**Sample text**

1 38 3. 11. More Projections Consider for a moment a subvariety Xc Y x P' where Y is an affine variety. Such a variety may be given as the zero locus of polynomials F(Wo , W1 ) homogeneous in the coordinates Wo , W1 on P', whose coefficients are regular functions on Y. The technique of elimination theory then tells us that the image Y of X under the projection map it,: Y x P l Y is the zero locus of the resultants of all pairs of such polynomials; in particular, it is a closed subset of Y. Next, suppose we have a subvariety X of Y x P2 and would like to make the same statement.

Tobe Generalize this to show that if (L i, j ) is any 2 x d matrix of linear forms on Pd such that for any (A i , 22) (0, 0) the linear forms {2. , d are independent, then the locus of [Z] E P d such that the matrix L i (Z) has rank 1 is a rational normal curve. 26. Other Rational Curves The maps vd involve choosing a basis for the space of homogeneous polynomials of degree d on P l . In fact, we can also choose any collection A0, , Am of linearly independent polynomials (without common zeros) and try to describe the image of the resulting map (if the polynomials we choose fail to be linearly independent, that just means the image will lie in a proper linear subspace of the target space Pm).

24. Verify the last statement. We can use this descri tion to see once again that there exists a unique rational normal curve through d + s oints in Pd no d + 1 of which are dependent. To do this, choose the subspaces Ai d-2 c= H be the span of the points , P, P. , for each value e P l , the hyperplanes Hi(2) intersec in a point p(A). The rational normal curve constructed in this way will necessarily. contain the d points Pi ; and given three additional points Pd +1, Pd + 2, and Pd+3 V116, can choose our parameterizations of the families of planes through the Ai so that the planes containing P - d-I-11 Pd+2, and Pd + 3 correspond to the values 2 = 0, 1, and CC E P l , respectively.

### Algebraic Geometry: A First Course by Joe Harris

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