Definition of a $\mathcal{O}(a,b)$?

Question

Can any one tell me what is the definition of this notation $\mathcal{O}(a,b)$.

I know $\mathcal{O}(a)= \widetilde{S}(a)$ for some ring $S$. Can $\mathcal{O}(a,b)$ be defined in the same way.

thanks in advance

Answer 1

If you look on page 26, you'll see that we're dealing with a product of projective spaces. I think this notation is the same as in Hartshorne II.6.6.1 and II.7.6.2- we're dealing with a line bundle which is of degree $a$ on the first factor and degree $b$ on the second factor. This would mean that $\mathcal{O}(a,b)=\widetilde{S_1(a)}\otimes \widetilde{S_2(b)}$ where $S_1=R[x_0,\cdots,x_m]$ and $S_2=R[x_0,\cdots,x_n]$ are the two rings corresponding to the two projective spaces.