Representability of a functor in the category of schemes

Question

I have read in some places that a functor of points of a scheme is representable if its defined by locally closed or open conditions. I would like to ask for some references about this fact. I don´t know exactly what closed or open conditions are.

Answer 1

Question: "I have read in some places that a functor of points of a scheme is representable if its defined by locally closed or open conditions. I would like to ask for some references about this fact. I don´t know exactly what closed or open conditions are."

Answer: If $F \in C:=Funct^{op}(Sch, Sets)$ is a functor we say $F$ is representable iff there is a scheme $X \in Sch$ and an "isomorpism of functors" $\eta: F \cong h_X$ where $h_X(S):=Mor_{Sch}(S,T)$ is the functor of points of $X$. Given a functor $F\in C$ we may speak of "open subfunctors" $U \subseteq F$ and an "open cover $U_i \subseteq F$" of $F$ by open subfunctors. There is the following criterium:

Theorem: A functor $F\in C$ is representable iff $F$ is a "sheaf in the Zariski topology" and there is an open cover $U_i$ of $F$ where $U_i \cong h_{X_i}$ are representable functors - represented by the schemes $X_i \in Sch$.

On the arXiv you find some papers of Vistoli on moduli functors and moduli spaces outlining this.