Algebraic Geometry: A First Course by Joe Harris

By Joe Harris

This publication is predicated 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. having said that, a number of phrases are so as concerning the reasons of the ebook. Algebraic geometry has built vastly during the last century. through 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 ideas for the main half have been grounded in geometric structures. This technique flourished in the course of the heart 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 20 th centuries. finally, the topic used to be driven past the boundaries of its foundations: by way of the tip of its interval the Italian university had advanced to the purpose the place the language and methods of the topic might not serve to precise or perform the information of its top practitioners.

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The basic observation is contained in the following exercise. 24. Let X c P" be any projective variety and 9: X -- Pin any regular map. Show that the graph F, = X x Pm OE P" x Pr" is a subvariety. Note that it is not the case that a map 9: X -- Pm is regular if and only if the a subvariety. 26 and denote by X = P 2 its image, the cuspidal curve Z0 Zi -= Z. , X ■ 2. 30 Regular Functions and Maps not regular, although its graph, being the same subset of X x P' as the graph of /2, is a subvariety of X x P 1 .

Observe in particular that if we restrict vd to a linear subspace it Pk Pn, we get just the Veronese map of degree d on 1Pk. F'or example, the images under the map v2 : P 2 –> P 5 of lines in P 2 give a family of conic plane curves on the Veronese surface S, with one such conic passing through any two points of S. More generally, we claim that the image of a variety Y c P" under the Veronese map is a subvariety of PA'. To see this, note first that homogeneous polynomials of degree k in the homogeneous coordinates Z on P N pull back to give (all) polynomials of degree d • k in the variables X.

Any four of them span P 3. 18, we will see that given any six points in P 3 in general position there is a unique twisted cubic containing all six. 13. Show that if seven points Pi'•, p 7 e P 3 lie on a twisted cubic, then the common zero locus of the quadratic polynomials vanishing at the pis that twisted cubic. 14. Rational Normal Curves These may be thought of as a generalization of twisted cubics; the rational normal curve C Pd is defined to be the image of the map vd : is IN; th, Pd given by I'd: [X0 X 1 ] , H Ex-g, xg-ix i , xn [Zo , zd].

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