Algebraic Geometry: A First Course by Joe Harris

By Joe Harris

This ebook 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 creation to the topic. in spite of this, a couple of phrases are so as concerning the reasons of the e-book. Algebraic geometry has built greatly during the last century. throughout the nineteenth century, the topic used to be practiced on a comparatively concrete, down-to-earth point; the most items of research have been projective kinds, and the thoughts for the main half have been grounded in geometric structures. This method flourished in the course of the center of the century and reached its fruits within the paintings of the Italian college round the finish of the nineteenth and the start of the 20 th centuries. finally, the topic used to be driven past the bounds of its foundations: via the top of its interval the Italian tuition had advanced to the purpose the place the language and strategies of the topic may well now not serve to specific or perform the guidelines of its most sensible practitioners.

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Example text

What this means is that given an arbitrary variety Z and a pair of maps a: Z --* X and fi: Z —t Y, there is a unique map cx x /3: Z —> X x Y whose compositions with the projections nx Xx Y and it } are a and fi, respectively. It is not \ft: hard to see that this is the case: to begin with, the map a x fl is certainly uniquely determined by a and A we X Y have to check simply that it is regular. This we do locally: say r0 G Z is any point, mapping via a and lq to points p e X and q e Y. Suppose p lies in the open set Z0 0 0 in P's, so that in a neighborhood of r0 the map a is given by A a: r i— [1, f 1 0, .

Already in the first lecture we have encountered on a number of occasions references to three basic notions in algebraic geometry: dimension, degree, and sm othness. We have referred to various varieties as curves and surfaces; we hive defined the degree of finite collections of points and hypersurfaces (and, implicitly, of the twisted cubic curve); and we have distinguished smooth conics from arbitrary ones. Clearly, these three ideas are fundamental to the subject;/they give structure and focus to our analysis of varieties.

20, the image of the product X xY P" x P'" is a locally closed subset of Pn""m , which we will take as the definition of "the product X x Y" as a variety. , the projection maps nx : X x Y —0 X and n y : X x Y Y are regular and the variety X x Y, together with these projection maps, satisfies the conditions for a product in the category of quasi-projective varieties and regular 29 Regular Maps maps. What this means is that given an arbitrary variety Z and a pair of maps a: Z --* X and fi: Z —t Y, there is a unique map cx x /3: Z —> X x Y whose compositions with the projections nx Xx Y and it } are a and fi, respectively.

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