algebraic geometry - a man was killed who tried to overhear

a scheme is not an algebraic curve because it may not be smooth over k.

one must add       principal ideal generated by x^p - a

a less sophisticated point of view is to view an algebraic curve as a coherent analytic sheaf

keep in mind  'enough injectives'


Weil conjectures
algebraic variety V defined over a finite field $\mathbb{F}_q $

(a famous mathematician, maybe Beilinson, who was listening, said   finding l-adic cohomology is a vital problem if ever there was one )

one can see an algebraic curve simply as a stack
but one must be careful to think about the fine moduli space

scheme whose points are in 1-1 correspondence with the coarse moduli space

if you understand all this  the Riemann hypothesis follows from Bombieri's proof of the Riemann hypothesis for function fields

one motivation for creating algebraic geometry was to prove the Riemann hypothesis but this part of history has been forgotten