page_title: Intersection theory
section_title: Intersection multiplicities
hierarchical_section_title: Intersection theory / Intersection theory in algebraic geometry / Intersection multiplicities
caption_attribution_description: parabola and lines intersecting to illustrate multiplicity two
context_page_description: In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.
context_section_description: The guiding principle in the definition of intersection multiplicities of cycles is continuity in a certain sense. Consider the following elementary example: the intersection of a parabola y = x² and an axis y = 0 should be 2 · (0, 0), because if one of the cycles moves (yet in an undefined sense), there are precisely two intersection points which both converge to (0, 0) when the cycles approach the depicted position. (The picture is misleading insofar as the apparently empty intersection of the parabola and the line y = −3 is empty, because only the real solutions of the equations are depicted).
  The first fully satisfactory definition of intersection multiplicities was given by Serre: Let the ambient variety X be smooth (or all local rings regular). Further let V and W be two (irreducible reduced closed) subvarieties, such that their intersection is proper. The construction is local, therefore the varieties may be represented by two ideals I and J in the coordinate ring of X. Let Z be an irreducible component of the set-theoretic intersection V ∩ W and z its generic point. The multiplicity of Z in the intersection product V · W is defined by
  ,
  the alternating sum over the length over the local ring of X in z of torsion groups of the factor rings corresponding to the subvarieties. This expression is sometimes referred to as Serre's Tor-formula.
  Remarks:
  The first summand, the length of
  is the "naive" guess of the multiplicity; however, as Serre shows, it is not sufficient.
  The sum is finite, because the regular local ring  has finite Tor-dimension.
  If the intersection of V and W is not proper, the above multiplicity will be zero. If it is proper, it is strictly positive. (Both statements are not obvious from the definition).
  Using a spectral sequence argument, it can be shown that μ(Z; V, W) = μ(Z; W, V).
caption: Intersection of lines and parabola
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