Surely a silly question, but anyway. Suppose I blow up a point P in the plane. Then the exceptional divisor E should have zero intersection with (the strict transform of) any curve in the plane, since they are all linearly equivalent to a line not passing through P. But the strict transform of a line L through P seems to be a line which intersects E transversely (if I've computed correctly) which would mean L.E=1. What have I done wrong?
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This just means that the pull-back process doesn't preserve linear equivalence of divisors. This would work if we had a flat morphism.
The exceptional divisor intersects trivially the total transform of a curve in the plane, not the strict transform.