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!

Positivity in Algebraic Geometry I, 5.2.B. – Karl Schwede May 28 '12 at 20:44