Symmetric algebra in the weighted projective setting

Question

Let $X \subset \mathbb{P}(V)$ be a hypersurface of degree $d$, where $V$ is a vector space. Then we have the $d$th part of the symmetric algebra $S^d(V^\vee)$.

Now suppose we have a degree $d$ weighted projective hypersurface. What would the analogue to $S^d(V^\vee)$ in this setting be?

Answer 1

The homogeneous coordinate algebra of the weighted projective space $\mathbb{P}(w_0,w_1,\dots,w_n)$ is the polynomial algebra $\Bbbk[x_0,x_1,\dots,x_n]$ graded by the rule $\deg(x_i) = w_i$.