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 truly is meant to be, because the identify indicates, a primary advent to the topic. nonetheless, a number of phrases are so as in regards to the reasons of the publication. Algebraic geometry has constructed enormously during the last century. through the nineteenth century, the topic was once practiced on a comparatively concrete, down-to-earth point; the most items of analysis have been projective kinds, and the ideas for the main half have been grounded in geometric structures. This procedure flourished in the course of the heart of the century and reached its fruits within the paintings of the Italian institution round the finish of the nineteenth and the start of the twentieth centuries. eventually, the topic used to be driven past the bounds of its foundations: via the top of its interval the Italian institution had advanced to the purpose the place the language and strategies of the topic may possibly now not serve to precise or perform the guidelines of its most sensible practitioners.

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**Sample text**

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).

If X pm is projective, we may specify a map of X to Pn by giving an (n + 1)-tuple of homogeneous pol-yriomials of the sialltç dep GG, as long as they are not simultaneously zero anywhere on X, this will determine a regular map. It happens, though, that this still does not suffice to describe all maps of projective varieties to projective space. As an example of this, consider the variety C c P 2 given by X2 + Y 2 — Z2, and the map (p of X to 1131 given by [X, Y, Z] The map may be thought of as a stereographic projection from the point p = [0, 1, 1]: it sends a point r c C (other than p itself) to the point of intersection of the axis (Y = 0) with the line The two polynomials X and Y — Z [X, Y — Z].

AX0 X? — Xn. The images C P 3 of these maps are called rational quartic curves in P 3. The following exercise is probably hard to do purely naively, but will be easier after reading the next lecture. 29. Show that Cao is indeed an algebraic variety, and that it may be described as the zero locus of one quadratic and two cubic polynomials. 19 that the curves C family of non-projectively equivalent curves. a continuously varying 16 1. 30. Varieties Defined Over Subfields of K This is not really an example as much as it is a warning about terminology.