On the initial ideal of a generic artinian Gorenstein algebra

In this note we show that the initial ideal of the annihilator ideal of a generic form is generated by the largest possible monomials in each degree. We also show that the initial ideal with respect to the degree reverse lexicographical ordering of the annihilator ideal of the complete symmetric form has this property, by determining a minimal Gröbner basis of it. Moreover, we determine the total Betti numbers for a class of strongly stable monomial ideals and show that these numbers agree with those for the degree reverse lexicographical initial ideals of the ideal generated by a sufficiently large number of generic forms, and of the annihilator ideal of a generic form.