Question about grade of an ideal

Question

Let $K$ be a field and $S=K[[X,Y,Z,W]]$. Consider the elements $f:=XW-YZ$, $g:=Y^3-X^2Z$ and $h:=Z^3-Y^2W$ of $S$ and set $R=S/\langle f\rangle$ and $I:=\langle f,g,h\rangle /f$. I know that $\mathrm{cd}(I,R)=1$. Is $\mathrm{grade}(I,R)=0$?

Answer 1

$R$ is a Cohen-Macaulay local ring and $$\operatorname{ht}I=\dim R-\dim R/I=\dim S/(f)-\dim S/(f,g,h)=3-2=1,$$ so the answer to your question is negative.