I have just started a module on Algebraic Curves. From what I understand part of the reason why projective spaces exist is to make sure Bézout's theorem holds for any curve.
This is what I mean: Bézout's Theorem says that the number of points of intersection between any two curves is equal to the product of their degree.
Now when we work in an affine space this does not always hold. Take $\mathbb{R}^{2}$ and let $f=ax+b$ and $g=cx+d$, then by Bézout's $f$ and $g$ have one points of intersection (by taking the product of the degrees since they are both of degree one). Suppose $a=c$ then $f$ and $g$ are parallel lines which we know do not intersect. Hence we get no point of intersection, i.e. Bézout's doesn't hold in this setting.
On the other hand it is always true for projective planes, take $\mathbb{P}^{2}(\mathbb{R})$ then two parallel lines will meet at a point at infinity.
This however only holds for the projective plane but not other projective spaces. For example take $\mathbb{P}^{3}(\mathbb{F})$ and consider the projective lines
$\{[X,Y,0,0]:[X,Y] \in \mathbb{P} \}$
$\{[0,0,Z,W]:[Z,W] \in \mathbb{P} \}$
Then these two do not intersect at any point.
Now given the above;
Under what conditions does Bezout's theorem (whenever $n≠2$) hold?
Could we not construct the projective spaces $\mathbb{P}^{n>2}(\mathbb{F})$ such that they behave like the projective plane (making sure all lines intersect once)?
Any help is appreciated!
The analog of Bezout's theorem in higher dimension is:
It is not clear to me what properties you would like to preserve in a projective space $\Bbb{P}^n$ with $n>2$ where every pair of lines meets in a a point. If you preserve reasonable properties, such as each pair of distinct points defining a unique line, and every three points being contained in a projective plane (in the usual sense of projective plane), then this is impossible.