Assume $X$ is a subscheme of $Y$ and $X,Y$ are irreducible. Then every irreducible component of the normal cone $C_{X/Y}$ dominates $X$?
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2$\begingroup$ That is not true. Let $A$ be affine $3$-space, $\text{Spec}\ k[s,t,u]$. Let $Y$ be the singular quadric surface, $\text{Spec}\ k[s,t,u]/\langle su-t^2\rangle$. Let $X$ be a line in the surface, $\text{Spec}\ k[s,t,u]/\langle s,t \rangle$. Then the normal cone $C_{X/\mathbb{A}^3}$ is the $\mathbb{A}^2$-bundle over the line $\text{Spec}\ (k[s,t,u]/\langle s,t \rangle)[S,T]$. The closed subscheme $C_{X/Y}$ is $\text{Spec}\ (k[s,t,u]/\langle s,t \rangle)[S,T]/\langle uS \rangle$. So one component, $\text{Zero}(S)$, dominates, but the other component, $\text{Zero}(u)$, does not dominate. $\endgroup$– Jason StarrCommented Mar 29, 2017 at 9:11
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$\begingroup$ Thank you very much!! Another question. Is there any good theorem, paper, book about how irreducible components of normal cone behaves? Sorry for my question is not precise. $\endgroup$– keatonCommented Mar 29, 2017 at 9:29
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$\begingroup$ Fulton's book "Intersection Theory" has some discussion of these issues. I do not know a definitive reference for irreducible components of the normal cone. $\endgroup$– Jason StarrCommented Mar 29, 2017 at 9:59
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$\begingroup$ The only easy statement seems to be that $C_{X/Y}$ is set-theoretically equidimensional, and connected in codimension $1$, being a flat degeneration of $Y$. $\endgroup$– Allen KnutsonCommented Apr 1, 2017 at 9:58
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