Let $Y\subseteq \mathbb{P}^n(k)$ be a quasi-projective variety. By Görtz, Wedhorn (page 32, Proposition 1.65) in order to show that $$h:Y\to \mathbb{P}^m(k), y\mapsto (f_0(y):\dots :f_m(y))$$ is a morphism of prevarieties where $f_i\in k[X_0,\dots,X_n]$ are homogeneous polynomials of the same degree which don't vanish simultaneously it suffices to show that each component of the restriction $h_{\mid h^{-1}(U_j)}$ is a morphism of prevarieties.
My questions:
- Why does that suffice?
- Where do we use that $Y$ is an open subset of a closed subset of $\mathbb{P}^n(k)$?
My thoughts:
By the definition of a morphism of prevarieties we have to show:
- $h$ is continuous
- for all open $U \subseteq \mathbb{P}^m(k)$ and all $f\in \mathcal{O}_{\mathbb{P}^m(k)}(U)$ we have $f\circ h_{\mid h^{-1}(U)}\in \mathcal{O}_Y(h^{-1}(U))$.
Maybe in the last point we can restrict ourselves to $U_i$ by Proposition 1.44 and the glueing property. But still we have to show something different than stated in Görtz: $$f\circ h_{\mid h^{-1}(U_i)}\in \mathcal{O}_Y(h^{-1}(U_i))$$