There are many instances in which theory over $\mathbb{C}$ is cleaner than theory over $\mathbb{R}$. For example, continuously differentiable functions over $\mathbb{R}$ are not necessarily twice differentiable, whereas entire functions over $\mathbb{C}$ are infinitely differentiable with a convergent power series. Also, $\mathbb{C}$ is algebraically closed, and so systems of polynomial equations over $\mathbb{C}$ can by analyzed with Hilbert's Nullstellensatz, whereas the real case requires the more complicated Stengle's Positivstellensatz.

In my experience, there seems to be greater ease in working in projective geometry instead of affine geometry, much like this ease with $\mathbb{C}$ instead of $\mathbb{R}$. For example, in projective space, any two distinct lines intersect at exactly one point, whereas in affine space, it depends on whether the lines are parallel.

**What are some examples in which theory over projective space is cleaner than theory over affine space?**

I am interested in a wide spectrum of examples, i.e., elementary/deep examples from continuous/discrete spaces involving algrebra/geometry/combinatorics. Presumably, a big list of answers will illustrate the distinguishing characteristics of projective spaces that make them so nice to work with in so many areas.

any collineation comes from a semi-linear automorphism, may be stated for affine spaces but in a clumsier form. $\endgroup$1more comment