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We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, calculating the multiplicity at a point is rather hard (in comparison to the affine case at least). So could someone give me a simple (but not completely trivial) example of $1$ or $2$ curves and calculate multiplicities, tangents and/or the intersection of them. Thank you in advance!

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