Here are some of the exercise questions which I am stuck:
Question 1: Give an example of a real hypersurface $M\subset\mathbb{C}^n$ such that $0\in M$, such that $M$ has a polynomial defining equation whose Hessian eigenvalues (1,1,-1) holds at every point; and such that there are complex lines through $0$ and lying in $M$. Then give a second example where $0\in M$, the Hessian has eigenvalue (1,1,-1) at every point, and $M$ is strongly pseudoconvex at $0$.
For the first part I could just simply take $r(z,\overline{z})=|z_1|^2+|z_2|^2-|z_3|^3$. But I can't solve the second part of the question.
Question 2: Give an example of a hypersurface $M$ in $\mathbb{C}^n$ containing $p$ for which there are no non-singular holomorphic curves through $p$ lying in $M$ but for which there are such no-constant singular holomorphic curves.
Does singular here means all the first derivatives are zero? What could be the example?