# Computing a tangent cone

In Arbarello–Cornalba–Griffiths–Harris' "Geometry of algebraic curves", Vol. 1, Chap. 2, the computation of the tangent cone in the following situation is explained:

Let $$f:X\rightarrow Y$$ be a morphism between smooth (complex) varieties such that $$X$$ is birational to $$f(X)=Z$$. Let $$p\in Z$$ such that $$f^{-1}(p)$$ (scheme-theoretic fiber) is smooth. Then the projectivized tangent cone $$\mathbb P(\mathcal C_{(Z,Y),p})$$ is computed.

Now, let us consider the following situation:
Let $$f:X\rightarrow Y$$ be a morphism between smooth (complex) varieties such that $$X$$ is birational to $$Z=f(X)$$. Assume that there are non-empty subsets $$Z_1\subset Z_2\subset Z$$ such that the fiber over a point of $$Z_2\backslash Z_1$$ is two reduced (smooth) points and the fiber over a point of $$Z_1$$ is one (thus double) point.

Is there a reference where the computation of $$\mathbb P(\mathcal C_{(Z,Y),p})$$ for $$p\in Z_1$$ is explained?