Riemann surfaces beginners question: (I am learning about
normalization of algebraic curves for the first time using Fulton's Algebraic topology and was doing fine until i hit a this snag)
SHORT FORM OF QUESTION: Would somebody who has a copy of Fulton's Algebraic Topology handy please open it to page 279 at the top and read the first paragraph of the proof of Lemma 20.3 and tell me if they see a typo in the line that says "take S to be the set where a_0(z)*d(z) is not equal to 0"? Specifically should it say "equal to 0" instead of "not equal to 0"? If somebody will assure me that this is simply a typo in the book there will be no need to concern ourselves with the long form. Thanks.
LONG FORM OF QUESTON: In Fulton on p 279 near the top he is using the discriminant of a polynomial F(Z,W) to show that outside a finite point set S an algebraic curve can be projected onto the complex plane as an n-sheeted covering.
He defines, for the degree n polynomial F(Z,W), the polyns a_0(Z) and d(Z) as the high order coeff of W and the dis- criminant, respectively, and then uses these to identify the finite set S of points to be deleted from C (the complex numbers not the curve -- he uses "C" for but but i can't). The goal is that the remaining points should be guaranteed to yield n distinct roots for F.
He says that the points z where we do have this guarantee are those for which both a_0(Z) and d(Z) are non-zero. So one would expect the set S of "undesirable points" to consist of those where one or both of a_0(Z) and d(Z) is zero.
So then why does he say "take S to be the set of z in C where a_0(z)d(z) is not equal to 0" (i am using "" for multiplication; he uses a dot). Should he not instead have said "take S to be the set of z in C where a_0(z)*d(z) is equal to 0"?
Sorry for this probably trivial question but everything is so new... i am unsure of my thinking. If somebody know- ledgeable could just say "yes of course it a typo" i would be happy and could proceed. Thanks.
PS: it would be nice if there were errata for math books generally because altho most typo's are easy to get past a cleverly planted one can hold you up for quite a while!