Let $X=V(P_1,\dots, P_r)\subset \mathbb P^{n+1}_k$ be a closed algebraic subset defined by polynomials of degree resp. $d_1,\dots,d_r$ such that $d_1+\cdots+d_r<n+1$. Is there a way to compute the cohomology groups $H^i(X,\mathcal O_X)$ of $X$ for $i\geq 1$?
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