Families of Artinian and one-dimensional algebras

Abstract

The purpose of this paper is to study families of Artinian or one-dimensional quotients of a polynomial ring R with a special look to level algebras. Let GradAlg H ( R ) be the scheme parametrizing graded quotients of R with Hilbert function H. Let B → A be any graded surjection of quotients of R with Hilbert function H B = ( 1 , h 1 , … , h j , … ) and H A , respectively. If dim A = 0 (respectively dim A = depth A = 1 ) and A is a “truncation” of B in the sense that H A = ( 1 , h 1 , … , h j − 1 , α , 0 , 0 , … ) (respectively H A = ( 1 , h 1 , … , h j − 1 , α , α , α , … ) ) for some α ⩽ h j , then we show there is a close relationship between GradAlg H A ( R ) and GradAlg H B ( R ) concerning e.g. smoothness and dimension at the points ( A ) and ( B ) , respectively, provided B is a complete intersection or provided the Castelnuovo–Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to “non-truncated” Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of Cohen–Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg H ( R ) , H = ( 1 , 3 , 6 , 10 , 14 , 10 , 6 , 2 ) , whose general elements are Artinian level algebras of type 2.