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In the proof of nonvanishing theorem, people used this concept, but I cant see its definiton.

The usual definition can be found in books which only deal with two dimensional case. I never see a definition on higher dimension. It seems that there are at least three descriptions concerned with multiplicity.

Let $D$ be an effective Cartier divisor on a variety $X$, let $P\in D$. Pick a local equation $f$ of $D$.

(1) Indeicating by Hartshorne exercise V.3.4, one can define the mulitiplicity of the Noetherian local ring $\mathscr{O}_{X,P}/f$ by Hilbert-Samuel polynomial.

(2) One can define $\operatorname{mult}_P D:=\operatorname{ord}_P(f)=\max\{n\in\mathbb{N};\ f\in \mathfrak{m}^n\}$.

(3) Set $\operatorname{mult}_PD=\min\{C.D;\ C \mbox{ is a curve through P}\}$.

Does these three description coincides? Where can I find a detailed treatment of basic notions like these? Thank you!

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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. –  Parsa May 28 '12 at 12:58
    
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. –  Karl Schwede May 28 '12 at 20:44
    
Thanks to the above answers! –  MZWang May 29 '12 at 0:32
1  
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. –  quim Dec 5 '12 at 10:18
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