 
Summary: Vector bundles in Algebraic Geometry
Enrique Arrondo
Notes(*) prepared for the First Summer School on Complex Geometry
(Villarrica, Chile 79 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 welldefined 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
