Non-level semi-standard graded Cohen-Macaulay domain with $h$-vector $(h_0,h_1,h_2)$

Abstract: Let $k$ be an algebraically closed field of characteristic 0, and $A$ a Cohen-Macaulay graded domain with $A_0=k$. If $A$ is semi-standard graded (i.e., $A$ is finitely generated as a $k[A_1]$-module), it has the $h$-vector $(h_0, h_1, ..., h_s)$, which encodes the Hilbert function of $A$. From now on, assume that $s=2$. It is known that if $A$ is standard graded (i.e., $A=k[A_1]$), then $A$ is level. We will show that, in the semi-standard case, if $A$ is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers $h$ and $n$, there is a non-level $A$ with the $h$-vector $(1, h, (h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).