Consider a degree $6$ weighted projective hypersurface $Y$ in $w\mathbb{P} := \mathbb{P}(1,1,1,2,3)$ (it's an index $2$ Fano threefold of degree $1$ in regular projective space). Let $H$ be the ample line bundle on $Y$ which gives an embedding of $Y$ into regular projective space.
I'm interested in computing the cohomology groups $H^i(Y, \mathcal{O}_Y(H))$. If this were, say, a cubic threefold (degree $3$ hypersurface in $\mathbb{P}^4$), then I'd use the short exact sequence $ 0 \to \mathcal{O}_{\mathbb{P}^4}(-1) \to \mathcal{O}_{\mathbb{P}^4}(1) \to \mathcal{O}_Y(H) \to 0 $ to compute the analogous cohomology groups.
Since my current case is a weighted projective hypersurface though, I'm not entirely sure of a way to proceed. The short exact sequence I mention above is a twist of the ideal sheaf sequence $0 \to I_Y = \mathcal{O}_{\mathbb{P}^4}(-Y) \to \mathcal{O}_{\mathbb{P}^4} \to \mathcal{O}_{Y} \to 0$ so I was wondering if there is an analogous ideal sheaf sequence for weighted projective spaces?
Another thought was to use the weighted projective Euler exact sequence: it would look something like $$ 0 \to \Omega_{w\mathbb{P}}^1 \to \bigoplus_{i=0}^4 \mathcal{O}_{w \mathbb{P}}(-D_i) \to \mathcal{O}_{w\mathbb{P}} \to 0 $$
where we would replace $D_i$ by the weights of $w \mathbb{P}$ when doing the computation. I suppose the next step would be to twist this and restrict somehow (?), so that the last term becomes $\mathcal{O}_Y(H)$. I'm not sure if this, as well as computing the resulting cohomologies from the first two terms is feasible though?
Many thanks.
Best Answer
It is more convenient to think here of $w\mathbb{P}$ as of a quotient stack; anyway, if $Y$ i smooth it does not pass through the stacky points $$ (0,0,0,1,0), (0,0,0,0,1) \in w\mathbb{P} $$ and therefore the stacky structure of $w\mathbb{P}$ plays no role for $Y$.
The advantage of $w\mathbb{P} = \mathbb{P}(w_0,w_1,\dots,w_n)$ as of a stack is that it comes with the line bundle sequence $\mathcal{O}(i)$ such that $$ H^p(w\mathbb{P}, \mathcal{O}(i)) = \begin{cases} A_i, & \text{if $i \ge 0$ and $p = 0$},\\ A_{-w-i}, & \text{if $i \le -w$ and $p = n$},\\ 0, & \text{otherwise}, \end{cases} $$ where $A = \Bbbk[x_0,x_1,\dots,x_n]$ with $\deg(x_k) = w_k$, and $w = w_0 + w_1 + \dots + w_n$. Now you can compute cohomology of restrictions of these line bundles to arbitrary hypersurfaces in the same way as in the usual projective space using the Koszul resolution $$ 0 \to \mathcal{O}(-d) \to \mathcal{O} \to \mathcal{O}_Y \to 0 $$ for a hypersurface $Y$ of degree $d$.