Motivation to do Algebraic Geometry

Hi,

there are many amazing examples of the use of algebraic topology for a
undergrad math student. For example the topological statement that
there are two points on the surface of the earth with same temperature
and same pressure has always been a motivating example to me to study
algebraic topology. There are many more of this kind like the
fundamental theorem of algebra etc. which beauty can be seen by non-
mathematicians as well.

Since a few month I try to understand things in algebraic geometry and
I'm searching for a similar motivation to do algebraic topology as the
above ones for algebraic topology. All the things I know so far which
demonstrate the use of algebraic geometry are, well, a little
complicated: I.e. the theorem of Fermat which has been proved by Weil
using algebraic geometry will be out of reach for me for many many
years (I hope one day I will understand something like that). So I am
searching for an illuminating easy example to demonstrate the use of
algebraic geometry methods, which can be a motivation for me to study
such things and which I will be able to understand in, say, one year
or so. Perhaps there are some examples in number theory?

Does anybody know examples like that?

Yours
Alex
.

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