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We investigate Borel ideals on the Hilbert scheme components ofarithmetically Cohen-Macaulay (ACM) codimension two schemes in P^n. We give abasic necessary criterion for a Borel ideal to be on such a component. Thenconsidering ACM curves in P^3 on a quadric we compute in several examples allthe Borel ideals on their Hilbert scheme component. Based on this we conjecturewhich Borel ideals are on such a component, and for a range of Borel ideals weprove that they are on the component.

Borel Degenerations of Arithmetically Cohen-Macaulay curves in P^3