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I'm reading “Graph on Surfaces” by Lando and Zvonkin, and I meet some problems about algebraic geometry.

The book defines a polynomial mapping(i.e. morphisms between affine spaces): $f:\mathbb{C}^n \to \mathbb{C}^n$ to be quasi-finite if for every $p \in \mathbb{C}^n, f^{-1}(p)$ is a finite set. And define the degree of a finite map to be the number of preimage of a "generic point"(not the generic point in scheme theory, but means "general points") in $\mathbb{C}^n$.

My first question is

  • Why is this definition reasonable? Explicitly, why there're generic points(which may refer to a dense open subset of $\mathbb{C}^n$ I guess) whose number of preimage are the same?

The book then states that consider polynomial mapping $f=(f_1,\cdots,f_n):\mathbb{C}^n \to \mathbb{C}^n$, such that each $f_i$ is a homogenous polynomial of degree $d_i$. Then consider all such polynomials, which form a vector space. Then the mappings which are not quasi-finite form a subvariety of this vector space. And my second question is

  • Why is it true?
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    $\begingroup$ First, the definition of finite is not quite standard, the above would be termed as quasi-finite and there is a major difference between that and the usual definition of finiteness. Your first question has a positive answer and it requires a proof. I suggest you look up a book in commutative algebra/algebraic geometry. The second question also requires a few steps if you do not know how to use incidence varieties. $\endgroup$ – Mohan Jun 12 at 16:05
  • $\begingroup$ Thank you, so could you please provide some reference? Thanks a lot! $\endgroup$ – Focus Jun 12 at 16:07
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    $\begingroup$ For the first question, Eisenbud's book is a good place. Second, I suspect msot of the books will prove something more general and not this specific case. $\endgroup$ – Mohan Jun 12 at 18:08

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