Let $R=\mathbb R[x,y,z]$ be the ring of polynomials in three variables, and let $l_1,..., l_n$, $n>1$, be a set of linear forms whose locus meet at the point $(1:0:0)$ in the projective space.
The question is how to compute the projective dimension of the $R$-module $R/I$ where $I=\left<l_1^r,...,l_n^r\right>$.
I know from the paper: Schenck H., Stillman M., Local cohomology of bivariate splines, that this dimension is 2, the reason as said in the paper is that: First we can change the coordinates to become in the two variable space. Secondly, we have at least two lines.