Doubt about the definition of fine moduli space

Question

I’m studying “Introduction to differentiable stacks” (Grégory Ginot) and I don’t understand a technicality in the (somewhat informal) definition of fine moduli space given at page 12.

Basically if we want to classify a certain class of algebro-geometric objects $\mathcal{G}$ up to isomorphism, a fine moduli space is a space $M$ whose points are in 1 to 1 correspondence with isomorphism classes of $\mathcal{G}$, and a map $\mathscr{U}\to M$ such that the fiber of $m$ is in the isomorphism class represented by $m$. This map $\mathscr{U}\to M$ should also be universal in the sense that any other family $\mathscr{F}\to N$ should be the “pullback of the universal family through an unique arrow $f:N\to M$”.

My source says that the phrase in italics means that the following diagram is cartesian: $\hspace{4 cm}$

What is the top arrow? Is it unique or are we requiring that $f$ is the unique arrow such that a top arrow that makes the diagram cartesian exists?

Answer 1

The correct statement is as follows. For any $F→N$ there is a unique arrow $f$ such that there is an isomorphism $F→N ⨯_M U$ over $N$.

Answer 2

Each fiber of $\mathscr U \to M$ is the object which is represented by its base point $m \in M$. In the same way, the family $\mathscr F \to N$ associates to points $n \in N$ its fiber, and in that way an element $\mathscr F_n$ of $\mathcal G$. Now the map $\mathscr F \to \mathscr U$ identifies $\mathscr F_n$ with $\mathscr U_m$, where $f(n) = m$.

I think an example can be helpful. As you now, a point $[v]$ in projective space $\mathbb P^n_{\mathbb C}$ represents a line $\mathbb Cv \subset \mathbb C^{n+1}$. Taking all those lines together, we obtain a complex¹ line bundle $$L = \{([v], w) : w \in \mathbb C v\} \subset \mathbb P^{n} \times \mathbb C^{n+1}.$$ As it turns out, $\mathbb P^n$ actually is a fine moduli space for this situation. So given any manifold $X$, together with a complex line subbundle $$L' \subset X \times \mathbb C^{n+1},$$ there is a unique map $$X \to \mathbb P^n$$ which sends $x \in X$ to the point $[L'_x] \in \mathbb P^{n}$. This map extends to a commutative diagram $$\require{AMScd} \begin{CD} L' @>>> L \\ @Vi'VV @ViVV\\ X \times \mathbb C^{n+1} @>f\times \operatorname{id}>> \mathbb P^n \times \mathbb C^{n+1}\\ @V\operatorname{pr}_XVV @V\operatorname{pr}_{\mathbb P^n}VV\\\ X @>>> \mathbb P^n \end{CD}$$ where $i'$ and $i$ are the given inclusions.

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¹ I guess all of this can also be done for real projective space, but I'm just more comfortable with complex projective space.