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Smoothness in Algebraic Geography
Author(s) -
Bruyn Lieven Le,
Reichstein Zinovy
Publication year - 1999
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611599011880
Subject(s) - mathematics , subvariety , moduli space , isomorphism (crystallography) , vector bundle , structure constants , algebraic structure , invariant (physics) , pure mathematics , combinatorics , algebra over a field , mathematical physics , crystal structure , chemistry , statistics , variety (cybernetics) , crystallography
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.