Two projective plane curves of degrees and with no common irreducible component have total intersection multiplicity . In particular, two nonempty projective plane curves always intersect.
Bézout's theorem is a fundamental result in algebraic geometry that concerns the intersection of projective curves. Specifically, it states that for two projective curves defined by polynomial equations in a projective space, the number of intersection points of these two curves, counted with multiplicities, is equal to the product of their degrees, provided that the curves intersect transversely (meaning they do not have singularities or tangential intersections at the points).
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