Let $S$ be a del Pezzo surface $S$ of degree $4$. There is an exact sequence
$ 0\to H^0(\mathbb{P}^4,I_S(2)) \to H^0(\mathbb{P}^4,\mathcal{O}(2))\to H^0(S,\mathcal{O}_S(2))\to0$ where $I_S$ is the ideal sheaf of $S$. Let $K$ be a canonical divisor. We have $H^0(S,\mathcal{O}(-2K_S))=13.$
The claim is $H^0(S,\mathcal{O}(-2K_S))=H^0(S,\mathcal{O}(2))$ and from this it follows that $S$ is the base locus of a pencil of quadrics. I neither understand the isomorphism nor how the statement about the base locus follows, could anyone elaborate?
This is from Dolgachev:" Classical Algebraic Geometry: a Modern View" (Thm 8.2.6)
Thanks a lot.
PS: a base locus of a pencil of quadrics is the intersection of two quadrics.