Let $X$ be the blow-up of $\mathbb{P}^2$ at three general points $p\_1,p\_2,p\_3$, that is a del Pezzo surface of degree six, and let $\pi\_i:X\rightarrow\mathbb{P}^1$ be the morphism induced by the projection from $p\_i$. 


Does anyone know a reference for the following classical fact?


Any morphism $f:X\rightarrow\mathbb{P}^1$ factors through $\pi\_i$ for some $i = 1,2,3$.



Pencils on del Pezzo surfaces
Let me sketch a proof. Let $D$ be the divisor class giving the map $f$. Then $D^2 = 0$ (because $D$ is the pullback of a point on $\mathbb{P}^1$). By adjunction formula $\deg(K\_D) = K\_X \cdot D$ is negative since $-K\_X$ is ample. Therefore, general fiber of $f$ is a rational curve, hence $K\_X \cdot D = -2$, hence $D$ is a conic (with respect to the anticanonical embedding of $X$). This means that $D$ is linearly equivalent to one of $C\_i$, hence $f$ coincides with one of $\pi\_i$.


