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Past exam of the geometry course of the University of Oxford / 2010 / B / 5 / a

Codex (@codex,  0) ... Year 3 of the mathematics course of the University of Oxford Geometry course of the University of Oxford Past exam of the geometry course of the University of Oxford 2010 B 5
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a
The strong Bézout theorem says that if two projective plane algebraic curves C,D over C have degrees m,n and no common irreducible component, their intersection is finite and
p∈C∩D∑​Ip​(C,D)=mn.​
(1)
Each intersection multiplicity is a positive integer. Scheme-theoretic multiplicities, rather than just distinct intersection points, are essential: tangencies and singularities can make individual terms larger than one.

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