Non-level semi-standard graded Cohen-Macaulay domain with h-vector (h0,h1,h2)

Let k be an algebraically closed field of characteristic 0, and A=⨁i∈NAi a Cohen-Macaulay graded domain with A0=k. If A is semi-standard graded (i.e., A is finitely generated as a k[A1]-module), it has the h-vector(h0,h1,…,hs), 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[A1]), then A is level. We will show that, in the semi-standard case, if A is not level, then h1+1 divides h2. 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).