Smoothness in Algebraic Geography

Let V be a vector space and let { e 1 ,…, e r } be a basis of V . An algebra structure on V is given by r 3 structure constants c i j h where e i ⋅ e j = ∑ h c i j h e h . We require this algebra structure to be associative with unit element e 1 . This limits the sets of structure constants ( c i j h ) to a subvariety of k r 3, which we denote by Alg r . Base changes in V (leaving e 1 fixed) give rise to the natural transport of structure action on Alg r ; isomorphism classes of r ‐dimensional algebras are in one‐to‐one correspondence with the orbits under this action. In this paper we classify the smooth closed subvarieties of Alg r which are invariant under the transport of structure action and study the singularities which may occur. In particular, we show that if r = n 2 then the closure of the locus corresponding to the matrix algebra M n ( k ) is not smooth for n ⩾ 3. This gives a negative answer to a question of Seshadri on the desingularization of moduli spaces of vector bundles over curves. 1991 Mathematics Subject Classification : 16G10, 16R30.