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I am looking for references for the following simple facts.

  1. Let $Y\subset \mathbb{P}^n$ be a variety (or pure-dimensional algebraic set). For $P\in Y$ denote by $e_p(Y)$ the (Samuel's) multiplicity of $Y$ at $P$. Then $\deg Y\geq e_p(Y)$.

  2. For simplicity, I will assume that $Y\subset K^n$ is affine variety. Suppose that for any $f\in I(Y)$ derivatives $\partial^a f(P)=0$ for all multi-indices $a=(a_1,\dots,a_n)$ satisfying $a_1/w_1+\cdots+a_n/w_n<1$. Then $e_p(Y)\geq \min w_{i_1} w_{i_2}\dots w_{i_d}$, where $d=\dim Y$.

I assume that the $K$ is algebraically closed field of characteristic 0.

I know how to prove these facts. For example, the first claim follows immediately from the Corollary 12.4 in Fulton's Intersection theory.

Added later: I was surprised to find a nice geometric description of the difference $\deg Y - e_p(Y)$ in the Appendix to Chapter 6 in Mumford's "Algebraic Geometry I: Complex Projective Varieties". I completely forgot about it.

The part 2 is false without some additional conditions on $Y$ (Cohen-Macaulay is sufficient).

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Hi Oleg, the comment of Steven Landsburg on this question:… implies that Samuel's book should have a reference to at least the first question of yours. –  auniket May 6 '13 at 19:56

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