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Summary: Vector bundles in Algebraic Geometry
Enrique Arrondo
Notes(*) prepared for the First Summer School on Complex Geometry
(Villarrica, Chile 7-9 December 2010)
1. The notion of vector bundle
In affine geometry, affine varieties are defined by zeros of polynomials in the sense that
a polynomial determines a well-defined function that we can evaluate and decide whether
it is zero or not. In projective geometry, however, a homogeneous polynomial of positive
degree never defines a function, and it only makes sense to say whether the polynomial
vanishes or not at some point. Moreover, the only regular functions on irreducible pro-
jective sets are the constants. We want to introduce the language of vector bundles as a
method to describe equations of projective sets, generalizing the notion of regular function.
We start with a couple of examples that illustrate what we have in mind.
Example 1.1. Let F K[x0, . . . , xn] be a homogeneous polynomial of degree d. If we
consider fi, the dehomogenization of F with respect of the variable Xi, now fi becomes a
function on the affine open set Ui = {xi = 0}. Let us try to analyze why it is not possible
to glue all these functions Fi to get a function on Pn
. If we write the coordinates in Ui as
x0
xi
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