A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it together, hope that somebody can help me:


Let $X$ be a projective scheme, say a surface over $\mathbb{C}$, let $S=Spec(R)$ be a parameter scheme. Assume we have a familiy $\mathcal{F}$ of coherent sheaves on $X\times S$, flat over $S$, let $\pi: X\times S \rightarrow S$ be the projection.


Now assume $V=\pi\_{\*}\mathcal{F}$ is locally free on $S$ and commutes with base change. We denote the total space of the vector bundle associated to $V$ by $T$, so $T=Spec(Sym(V^{\vee}))$, and denote by $\alpha: T \rightarrow S$ the projection.


Define $\beta$ to be the map $(id,\alpha): X\times T \rightarrow X\times S$.


Question: Does $\beta^{\*}\mathcal{F}$ have a canonical (or tautological) global section?


An element of $X\times T$ is a triple $(x,s,t)$ where $t\in H^0(X\_s,\mathcal{F}\_s)$. Now the fiber of $\beta^{\*}\mathcal{F}$ over $(x,s,t)$ should be just the fiber of $\mathcal{F}$ over $(x,s)$. So i could look at the image $t((x,s))$ of $t\_{(x,s)}$ in the fiber over $(x,s)$. 


Is $\sigma: X\times T \rightarrow \beta^{\*}\mathcal{F}$, $(x,s,t)\mapsto t((x,s))$ a well defined global section, that is: do we have $\sigma\in \Gamma(X\times T,\beta^{\*}\mathcal{F})$? Is this a canonical or tautological section? I mean, there is no other reasonable way to define the map $\sigma$. Does this make sense at all?



Canonical (tautological) section of a family of sheaves
What you have is the fiber diagram.


$$
 \begin{array}[c]{ccc}
X\times T & {\stackrel{\beta}{\rightarrow}} & X\times S \\
{\scriptstyle \delta} \downarrow & & \downarrow\scriptstyle{\pi}\\
T & {\stackrel{\alpha}{\rightarrow}} & S
\end{array}
$$


Now, there is a natural morphism
$\alpha^\* V = \alpha^\*\pi\_\*\mathcal{F}\to \delta\_\*\beta^\*\mathcal{F}$. Compose it with the canonical section $\mathcal{O}\_T\to \alpha^\*V$. By adjunction, you get a section $\mathcal{O}\_{X\times T}\to \beta^\*\mathcal{F}$.


