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It is (probably?) well-known that the birational isomorphism $\def\Q{\mathbb Q}\Q\to\{(x,y)\in\Q^2|x^2+y^2=1\}, t\mapsto(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})$ gives a characterization of all Pythagorean triples (over $\mathbb Z$). In my opinion this is a good example relating algebraic geometry to elementary number-theoretic applications (which can be understood by a wider audience). Even though one does not need to have a notion of birational isomorphism to complete the proof, I think this idea generalizes to scenarios where birational isomorphisms would be of help.

I am looking for examples similar in spirit to this one. My general requirements are:

  1. Embodies an idea from algebraic geometry. (As a start, maybe the first 2 chapters of Hartshorne? Though I would prefer examples using only knowledge of varieties.)
  2. The application is "easy" or "elementary" (e.g. elementary number theory). I wanted to say accessible to first-year undergraduates, but so many of them already know algebraic geometry nowadays, so I don't know...
  3. Hopefully it provides some motivation to study algebraic geometry. (Ideally, this should be a consequence of the above 2 points?)

If there is already a similar question here, I hope someone can point me to it, and I'll be happy to close/delete this one. Advice on improving the question is also welcome.

Tesla Daybreak
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    You might be interested by an old question of mine here. In the same vein, in many cases, determinants provide interesting varieties. Example : the Pfaffians. – Jean Marie Feb 23 '22 at 20:41
  • The assocation in your first sentence is the inverse of the stereographic projection $p$ from the north pole of $S^1$ to $\mathbb R$ which is a bijection (even a homeomorophism) . Clearly rational points of $S^1$ are mapped to rational points of $\mathbb R$. – Paul Frost Feb 23 '22 at 23:41

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