Denote by $\mathcal{O}_{p}$ the localization of $\mathbb{C}[x, y]$ at the ideal $(x, y)$.
Let $I$ be an ideal of $\mathcal{O}_{p}$ generated by two elements. Moreover require that $\mathcal{O}_{p} / I$ is finite dimensional algebra.
Consider a restriction of the ideal at a line, passing through the origin. It restricts as an ideal. Any ideal in the local ring of the origin of $\mathbb{A}^1$ is generated by $x^k$.
There are finitely many lines for which $k$ is greater than for general line. Let's call them special.
Question: Are there ideals for which there are more than one special line? Could you give an explicit example?
Example $I= (x^2, y^3)$. There is one special line $x=0$. For general line $k=2$. For special line $k=3$.
My question is answered by Mohan. I just want to write down the details.
$$I = \Big( x(x+y) , \ y^3 \Big)$$
Let's show that $\mathcal{O}_p / I$ is finite dimensional. Indeed $$y^2 \ x(x + y) - x \ y^3 = x^2 y^2 \in I$$ $$xy \ x(x+y) - x^2 y^2 = x^3 y \in I$$ $$x^2 \ x (x+y) - x^3 y = x^4 \in I$$
If $x^4, y^3 \in I$ then $\mathcal{O}_p / I$ is finite dimensional.
Consider a line $y = ax$. The restriction of $I$ is generated by $x^2(1+a)$ and $a^3 x^3$. So $y=-x$ is a special line. $x=0$ is another special line. So there are two special lines.