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First of all the definitions:

  • Let $k$ be an algebraically closed field. Then the affine space with dimension $n$ is defined as $\mathbb{A}_k^n := k^n$.

  • Furthermore, the projective space with dimension $n$ is defined as $\mathbb{P}_k^n := (\mathbb{A}_k^{n+1} \setminus \{0\})/\sim$ with $(x_0, \dots, x_n) \sim (y_0, \dots , y_n) :\Longleftrightarrow \text{there exists a } \lambda \neq 0 \text{ such that } y_i = \lambda x_i \text{ for all }i$.

In our lecture we have defined stuff like varieties, the Zariski topology, regular functions, their local rings etc. and have shown that these can be generalized to the projective space under certain conditions. My problem is that I have not understood yet why we need all this stuff and what things does the projective space provide which the affine space can not.

Could someone possibly explain this question as easily as possible? I have found this link but I find this one too difficult to understand. Maybe it would also help to give a explanation/interpretation of the other definitions like regular functions as well. Any help is very appreciated.

Diglett
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    I did an answer here about it : http://math.stackexchange.com/questions/1179312/why-are-projective-spaces-and-varieties-preferable/1179342#1179342 Basically, projective space is compact and everything behave better if the space is compact, for example for intersection theory this is important to work with projective varieties. –  Mar 29 '17 at 15:41

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