Degenerations of Monomial Ideals

We describe the degenerations of monomial ideals in $K[[x,y]]$ with ${\rm Aut}(K[[x,y]])$-orbit of dimension at most $3$. In particular, we determine the monomial ideals that any power of $(x,y^4)$ can degenerate to and make a conjecture about all the ideals that the powers of $(x,y^4)$ can degenerate to. We also give some numerical evidence linking the characteristics in which one ideal degenerates to another with the enumeration of lattice paths.

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Published 1 January 2004