I am getting very confused by all the equivalent definitions of regular functions, morphisms, rational functions and rational maps. I follow the terminology and definitions of Hartshorne, but in the end my aim is to understand the definitions of Silvermann in 'The arithmetic of elliptic curves', I.3.
- Is Lemma I.3.6 of Hartshorne also valid when $Y$ is a quasi-affine variety? I think it does, but I cannot see why he does not mention it.
- So any morphism to a (quasi?)-affine variety is of the form $f=(f_1,...,f_n)$ for regular functions $f_i$. Is there a similar characterization when $Y$ is any quasi-projective variety? I think we cannot just use $[f_0,...,f_n]$ as above, because by I.3.4 of Hartshorne all regular functions on a projective variety are constant, so this would imply that all morphisms from a projective variety are constant, which cannot be. However, Silvermann uses a similar description for rational maps, that are basically morphisms that are only defined on an open subset, so I guess a similar description for morphisms should be possible. Would it be correct, for instance, to say that morphisms correspond to functions of the form $f=[f_0,...,f_n]$ with the $f_i$ rational (instead of regular) functions, with the equivalence given by multiplication of all $f_i$ by the same rational function $g$?
- Is the equivalent characterization with homogeneous polynomials of the same degree given in Remark I.3.2 of Silvermann only valid for rational maps on projective varieties, or also for morphisms and rational maps on quasi-projective varieties?