tailieunhanh - Algebraic Geometry - KS. Kedlaya

In this lecture, I’ll give a bit of an overview of what we will be doing this semester, and in particular how it will differ from . We will start in earnest with the rudiments of category theory in the next lecture. Hopefully document content to meet the needs of learning, work effectively. | MIT OpenCourseWare http Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use visit http terms. Algebraic Geometry . Kedlaya MIT Spring 2009 Introduction In this lecture I ll give a bit of an overview of what we will be doing this semester and in particular how it will differ from . We will start in earnest with the rudiments of category theory in the next lecture. 1 Where we were and where we need to go In we studied the notion of an abstract algebraic variety over an algebraically closed field. This combines a lot of the commutative algebra developed in the early 20th century largely to explain the geometric reasoning of the masters of the Italian school with Weil s fundamental idea to glue affine algebraic varieties in the same way that one glues local charts together to build manifolds. So what s left We would like to deal with phenomena of nonreducedness for instance as it emerges under degenerations. One of the key ideas of the Italian school for understanding things like the geometry of the moduli space of curves was to notice that if you have a family of algebro-geometric objects defined in terms of a parameter t then the behavior of a particular member of the family is sometimes much simpler than that of a general member. For instance for a general t the elliptic curve y2 x3 tx 1 does not have a rational parametrization but it does in the special case t 0. One can often understand something about the general member of the family by first analyzing a special member then figuring out how the information you are looking for gets transmitted back to the general member via the degeneration. However degenerations of algebraic varieties are not always best viewed as algebraic varieties. For example if t 0 then the homogeneous polynomial y2 tx2 in x y z over say the complex numbers defines a pair of lines. The degeneration at t 0 however is the single line y 0 because .