I'm doing a module on algebraic curves which follows Fulton's book and I'm very confused.
Let K be a field. Right at the beginning of chapter 3 he defines an affine plane curve to be an equivalence class of non-constant polynomials in K[X,Y] which are related by a constant multiple to one another. So this means each curve would have a unique monic representative. Hence his definition in that chapter seems to define the curve to be the polynomial rather than the vanishing set.
However earlier in the book in chapter 1 he says a plane curve is the vanishing set of a non-constant polynomial in affine 2 space over K. This seems to be the definition I find over and over again wherever I go on the internet. I agree this is the more intuitive definition for a curve, but the problem is that so much that we define for curves in Fultons book seems to use the definition of it as the polynomial.
Let me illustrate my problem. X and X^2 are both valid polynomials in K[X,Y] and they both have the same vanishing set. But for X^2 I would conclude every point on its vanishing set is a multiple point (as both partial derivatives are zero) and for X I would conclude all points are simple points. So if I define the curve as the actual subset of affine 2 space I would conclude that whether a point is simple or not isn't well defined. You can say I could rectify this by saying the right one to do calculations with is X and not X^2 because it has the multiplicity, but Fulton's book doesn't seem to mention this at all.
The problem gets even worse for intersection numbers.