Construct an example: irreducible algebraic variety induces a reducible germ.

Question

This is another exercise of Daniel Huybrechts' book "Complex Geometry An Introduction". The question asks to construct examples of: 1. Reducible variety induces an irreducible germ. 2. Irreducible variety induces a reducible germ.

For #1, I found that a variety $X=\{0, 1\}$ of two single points is reducible while at either point it induces an irreducible germ.

For #2, can any one give an example?

Answer 1

The cusp $y^2=x^2+x^3$ in $\mathbb C^2$ is irreducible but has a reducible germ at the origin $(0,0)$.