Timeline for What is the multiplicity of a Cartier divisor at a point？

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Jul 21, 2017 at 19:26	history	bumped	CommunityBot		This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
Jun 21, 2017 at 18:45	answer	added	Tomo		timeline score: 2
Dec 5, 2012 at 10:18	comment	added	quim		In (3), the intersection product should be replaced by the local intersection multiplicity. Then, if X is smooth at P, the three definitions are equivalent.
Dec 5, 2012 at 8:29	history	edited	MZWang	CC BY-SA 3.0	added 756 characters in body
May 28, 2012 at 20:44	comment	added	Karl Schwede		Often its used as follows in the context you are considering. If $D$ is a Cartier divisor passing through a smooth point $x \in X$, then if $\pi : Y \to X$ is the blowup of $X$ at $x$ with exceptional divisor $E$, then $$\pi^D = D' + n E$$ where $D'$ is the strict transform of $D$ and $n$ is exactly the multiplicity that Parsa described above. Anyways, here's another reference: Lazarsfeld, Positivity in Algebraic Geometry I*, 5.2.B.
May 28, 2012 at 12:58	comment	added	Parsa		I imagine that if $f$ is a local equation for the divisor at a point $x$, then the multiplicity is the highest power of the maximal ideal $\mathfrak m_x$ that $f$ lies in.
May 28, 2012 at 12:45	history	asked	MZWang	CC BY-SA 3.0