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?