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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.

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    $\begingroup$ The fundamental theorem of projective geometry, any collineation comes from a semi-linear automorphism, may be stated for affine spaces but in a clumsier form. $\endgroup$
    – Leo Alonso
    Nov 15, 2017 at 15:52
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    $\begingroup$ @DustinGMixon : What level of answer are you looking for? Bézout's theorem comes to mind as an elementary example. $\endgroup$
    – Timothy Chow
    Nov 15, 2017 at 16:10
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    $\begingroup$ Over $\mathbf C$ say, projective space is compact, but affine space is not. There are numerous good properties that are essentially consequences of this: global holomorphic functions are constant, the image of closed subset under a holomorphic map is again closed, (fancier) cohomology groups of coherent sheaves are finite-dimensional vector spaces, and so on. $\endgroup$
    – Pooter
    Nov 15, 2017 at 16:22
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    $\begingroup$ This recent question on Math.SE may give another elementary example. It is a problem on Möbius transformations which looks daunting but turns into simple linear algebra in the framework of projective transformations of the complex projective line. $\endgroup$
    – Giuseppe Negro
    Nov 15, 2017 at 16:26
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    $\begingroup$ Regarding the real and complex numbers, the point is that the theory of real closed fields is not uncountably categorical unlike the theory of alg. closed fields of char 0. Now for projective spaces versus affine spaces, I don't know if some simplicity can be detected in the theory itself as in the previous case. Regarding examples, any intersection theory is likely to be ill-behaved in non projective stuff (or non proper stuff). $\endgroup$
    – user40276
    Nov 15, 2017 at 19:34

4 Answers 4

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From Timothy Chow:

Bézout's theorem comes to mind as an elementary example.

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From Leo Alonso:

The fundamental theorem of projective geometry, any collineation comes from a semi-linear automorphism, may be stated for affine spaces but in a clumsier form.

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From Pooter:

Over $\mathbf{C}$ say, projective space is compact, but affine space is not. There are numerous good properties that are essentially consequences of this: global holomorphic functions are constant, the image of closed subset under a holomorphic map is again closed, (fancier) cohomology groups of coherent sheaves are finite-dimensional vector spaces, and so on.

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From Francois Ziegler:

More automorphisms $\rightarrow$ simpler classification of conics or quadrics.

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