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 $\GradAlg^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$. Let $B \to A$ be any graded surjection of quotients of $R$ with Hilbert function $H_B$ and $H_A$, and h-vectors $h_B=(1,h_1,...,h_j,...)$ and $h_A$, respectively. If $\depth A = \dim A \leq 1$ and $A$ is a ``truncation'' of $B$ in the sense that $h_A=(1,h_1,...,h_{j-1},\alpha,0,0,...)$ for some $\alpha \leq h_j$, 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 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.