I am working with embeding using ample divisors, and I want to determine using the Hilbert series a model for a quartic surface. For that I wonder if:
There exists a program in which I provide as input a Hilbert series of a projective variety, and it determines if it is a complete intersection on a weighted projective space or not?
For example, I have computed for a certaint quartic smooth surface $X$ and a smooth curve $C\subset X$, the Hilbert series of the graded ring $\bigoplus_{n\geq 0}H^0(X,nC)$, it is given by $$P(t)=\sum_{n\geq 0}(1+2n^2)t^n=\frac{3t^2+1}{(1-t)^3},$$ but, I have not been able to determine whether this allows me to see $X$ as a complete intersection in a weigthed projective space. I think, that maybe exists some program which determine such a thing.
If anyone have any clue it will be welcome. Thanks!!