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What exercises should one solve (understanding proofs included) to gain an intuition for algebraic geometry?

What are examples of (not too hard) problems that algebraic geometry handles easier than elementary approaches?

Adiji
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  • Can you give us some sort of indication of your background? Would intuition for algebraic number theory, thinking about the theorems/notions as being geometric, be an answer to your problem? Do you know scheme theory? etc. – Alex Youcis Jul 23 '14 at 18:20
  • Why my question?: I can read texts about "abstract" algebraic geometry, understand the single steps, but don't see the geometry behind it. Also when it comes to concrete problems, I feel not comfortable. So I think I should DO some specific exercises or have some chosen examples in mind. But which ones? Geometric intuition for algebraic number theory would be great. And yes, I think I can say that I know scheme theory. – Adiji Jul 23 '14 at 19:48
  • Maybe this (http://math.stackexchange.com/questions/700641/is-there-a-geometric-interpretation-of-inert-primes/700951#700951) would be of interest to you. Another good example to see geometry is to intuit what various geometric morphisms mean. For example, what does 'etale' look like? How can you interpret the infinitesimal lifting property for smooth morphisms as saying 'tangent vectors lift'. Why do we have the exact seq. $f^\ast \Omega_{Y/Z}\to \Omega {X/Z}\to\Omega{X/Y}\to 0$, what geometrically does this mean. Why don't we have left exactness, and why does the lifting tangent vectors – Alex Youcis Jul 23 '14 at 20:00
  • property of smoothness tell us that if $f$ is smooth, we can add a $0$ to the left? How can you picture the Riemann-Hurwitz formula (perhaps taking intuition from the case of Riemann surfaces, and triangulations)? Why does $\pi_1\left(\mathbb{G}_m\right)=\widehat{\mathbb{Z}}$, and why does this make sense geometrically. Are these the type of things you're looking for? – Alex Youcis Jul 23 '14 at 20:02
  • They are! Thank you. But also important exercises one should do to understand geometric ideas behind definitions. E.g. the illustration you mention for (missing) left exactness. For the second question I would enjoy elementary problems that you can explain to non-algebraic geometers, that are "easy" to solve with alg. geo. or where geometric ideas help to understand the difficulty of the problem. – Adiji Jul 24 '14 at 07:27

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