Article:
Preprint, 12th July 1996.
We fix an algebraically closed groundfield k and a finite-dimensional k-algebra A. The variety modA(V) of all A-modules on the k-space V is a GL(V)-variety in a natural way, the orbits are exactly the isomorphism classes. By definition a module M degenerates to N iff the orbit closure of M contains N. There is a sufficient and a necessary criterion for degeneration of modules. While it is easy to see that adding direct summands does no harm, ie. if M degenerates to N then M+L degenerates to N+L, the inverse, ie. cancellation, is a difficult problem.
Here we consider the case of the commutative algebra with two generators a and b and relations a2 = 0 = b2. J. CARLSON pointed out that this algebra does not allow cancellation of direct summands in degenerations. Therefore the necessary condition cannot be sufficient, since it allows cancellation.
In this paper we give a detailed analysis of the degenerations of this algebra.