I am currently studying quotients of varieties, I use "Algebraic Geometry" by J. Harris. However, there are a few things I don't quite understand:
- (p. 123) At first it says that such a map $\pi$ is surjective, however in the second paragraph surjectivity is never mentioned. Is it not needed anymore? I found this Wikipedia article: https://en.wikipedia.org/wiki/Categorical_quotient It looks suspiciously similar to the definition of Harris, and in the Wikipedia article it specifically mentions that $\pi$ does not necessarily have to be surjective. However, in Harris book, a few pages later he actually shows that a certain quotient map $\pi$ is surjective. That confuses me.
- (p. 124) The sentence starts on the first picture. I don't understand, what would be stupid about composing these two maps?
- (p. 124) Here is an example where the quotient does not seem to exist. Here he also includes "surjective". But why does such a morphism not exist? Is it because the preimage of a closed subset has to be closed again under a morphism? In a variety with only two points $\{a, b\}$, where we identify $b = \mathbb{A} ^1 - \{0\}$, $b$ would be a closed point, however its preimage is open in $\mathbb{A} ^1$. Does this explanation work? Or maybe is because: A morphism would be a polynomial $p$, which needs to satisfy $p(0) = a$ and $p(x) = b$ for all $x \neq 0$. But such a polynomial does not exist.
Please tell me if I said something wrong, I am stuck on this topic for a while now and I really want to learn. Thanks in advance.