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  "body": "\n## Summary\n\nTwo subvarieties meet along the components of their intersection.\nThe intersection multiplicity along such a component is a positive integer that says how many times it should be counted.\nFor curves in the plane it is the dimension of a local ring, and in general it is given by Serre's Tor formula.\nWith these multiplicities, counts of intersection points become stable under deformation, which is what makes [[Bézout's theorem|KWNVF194D8]] and [[intersection theory|4HV4WTAX9P]] work.\n\n## Explanations\n\n### L1 human\n\nWhen a line crosses a curve transversally, the crossing point counts once.\nWhen the line is tangent to the curve, two crossings have merged into one point, and the point should count twice.\nThe intersection multiplicity makes this precise.\nFor the parabola $y = x^2$ and the line $y = 0$, the only common point is the origin, and its multiplicity is $2$.\nMoving the line to $y = \\epsilon$ with $\\epsilon \u003e 0$ splits it into the two points $x = \\pm\\sqrt{\\epsilon}$.\n",
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  "name": "Intersection multiplicity",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/GPQ/GPQF9GK4T9.md",
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