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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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    $\begingroup$ 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. $\endgroup$
    – Parsa
    May 28 '12 at 12:58
  • $\begingroup$ 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. $\endgroup$ May 28 '12 at 20:44
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    $\begingroup$ 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. $\endgroup$
    – quim
    Dec 5 '12 at 10:18
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See Fulton, intersection theory, section 4.3. more generally for a closed subscheme X of a pure dimensional scheme Y, and an irreducible component V of X, he defines the multiplicity of Y along X at V. When X and Y are varieties, this number is simply the coefficient of [X] in s(X,Y), the Segre class of the closed subscheme. When Y is an irreducible divisor and X is a codim 1 subvariety, blow up Y along X with projection p. Then $p_\ast$ (proper pushforward of cycles) of the exceptional divisor equals [X] times a number which is the multiplicity $e_XY$. When Y is not irreducible, just apply the above formula to each irreducible component and its intersection with X, then add up the numbers weighted by the geometric multiplicities of each component in Y.

More generally (e.g. when codim $n$ of X exceeds 1) you have the formula (X,Y varieties) $$e_XY[X]=(-1)^{n-1}p_\ast(\tilde X^n)$$ and you extend to reducible Y in the same way. Here $\tilde X^n$ denotes the nth self intersection of the exceptional divisor of the blowup of Y along X.

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