Families of Artinian and one-dimensional algebras

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 GradAlgH(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 HB=(1,h1,…,hj,…) and HA, respectively. If dimA=0 (respectively dimA=depthA=1) and A is a “truncation” of B in the sense that HA=(1,h1,…,hj−1,α,0,0,…) (respectively HA=(1,h1,…,hj−1,α,α,α,…)) for some α hj, then we show there is a close relationship between GradAlgHA(R) and GradAlgHB(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 GradAlgH(R), H=(1,3,6,10,14,10,6,2), whose general elements are Artinian level algebras of type 2.