Adjunction formula projective hypersurface interpretation

Question

I was reading a proof that all degree 4 hypersurfaces $X$ in $P^3(\mathbb{C})$ are K3 surfaces. When they were showing that the canonical bundle $\omega_X$ of $X$ is trivial, they suddenly used the fact that $\omega_X \cong \mathcal{O}_X(-n-1+d)$ and after some Googling, if found that this was an application of the adjunction formula. However, I'm having trouble understanding what actually the object $\mathcal{O}_X(-n-1+d)$ is well defined. Because this is the sheaf of holomorphic functions on $X$ but how can you define that on a natural number?

Answer 1

The notation $\mathcal{O}_X(n)$ (for any $n\in \mathbb{Z}$) does not mean the evaluation of $\mathcal{O}_X$ at $n$ (which does not make sense), but the $n$th Serre twist of $\mathcal{O}_X$.

Precisely, a projective variety has a "twisting sheaf" $\mathcal{O}_X(1)$, and for any sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules, its $n$th twist (for $n$ nonnegative) is $$\mathcal{F}(n) = \mathcal{F} \otimes \mathcal{O}_X(1)\otimes \dots \otimes \mathcal{O}_X(1)$$ When $n$ is negative you have to use $\mathcal{O}_X(-1)$, which is dual to $\mathcal{O}_X(1)$.