If I have two curves inside a projective surface, their intersection number should be the degree of their intersection product, i.e., I "move them" a bit so that they intersect properly and then consider the divisor consisting of their intersection points counted with the right multiplicity, and then add up those multiplicities. So far, is that right? I know this is a bit informal and I do not want formal answers based on formulas and computations, I just want an intuitive explanation: how is it possible that I ever get a negative number? Even more specifically, how can the multiplicity for the intersection of two curves at a point be negative? If they intersect properly, should the intersection number not just be 1 at every point, so that the total intersection number of the two curves is just the number of intersection points (which is clearly non-negative)?
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When you perturb it, it might fails to be a complex curve and thus the intersection no might be negative. – Arctic Char Jan 09 '20 at 17:01
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@ArcticChar Do you mean that "perturbing up to rational equivalence" is something I do on divisors, not on curves, so that I might end up with some formal sum of curves? But even then, should it not still be an effective divisor, hence only with positive coefficients? Or can an effective divisor be rationally equivalent to a non-effective divisor? – 57Jimmy Jan 09 '20 at 17:58