  Let $f_1, ..., f_n$ be homogeneous polynomials generating a generic ideal $I$
in the ring of polynomials in $n$ variables over an infinite field.
Moreno-Soc\'ias conjectured that for the graded reverse lexicographic term
ordering, the initial ideal ${\rm in}(I)$ is a weakly reverse lexicographic
ideal. This paper contains a new proof of Moreno-Soc{\'\i}as' conjecture for
the case $n=2$.
