December 13, 2009.
After the above article appeared, the following rather concise example over a field of characteristic zero was communicated to the author by Peter Schenzel. These cubics yield the same minimal free resolution as discussed in section 3 of the article.
f = x_{3}^{3} - x_{2} x_{3} x_{4} + x_{1} x_{4}^{2}
g = x_{1} x_{2} x_{3} - x_{0} x_{3}^{2} - x_{1}^{2} x_{4}
h = x_{1}^{3} - x_{0} x_{1} x_{2} + x_{0}^{2} x_{3}