Let $X$ be the hyper-surface defined by $$f:=\sum_{i=1}^k x_i^n=0$$ in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal $$I=(x_1^{n-1},\dots , x_k^{n-1}) $$ What is the multiplicity of $Y$ in $X$? Is it $n(n-1)^{k-1}$?
thanks
This follows from the discussion in Fulton's book on intersection theory, Example 4.3.5. However, your terminology isn't quite right. What you want is the multiplicity of $Y$ along $X$ at $0$, see Fulton, Example 4.3.4. Fulton quotes a result of Samuel that one can reduce this multiplicity to the multiplicity for an ideal $I' \subset I$ generated by a regular sequence $f_1, \ldots, f_{k - 1}$ where the $f_i$ are linear combinations of the generators you gave for $I$ (this is actually a bit surprising). Then $I'$ defines a complete intersection supported at $0$ given by homogeneous polynomials of degrees $n, n - 1, \ldots, n - 1$. A standard argument shows that here the number is indeed as you predict.
