Let $p : X \to S$ be a projective morphism between two Noetherian $\mathbb{C}$-schemes of finite type with connected fibres. Let $O_X(1)$ be a very ample line bundle on $X$ relative to $S$. Given a polynomial P there exists a projective relative moduli space $$M_{X/S}(P) \to S$$ which parametrizes families of semistable sheaves on the fibres for $p$ with Hilbert polynomial $P$ (see for instance the book of Huybrechts and Lehn ).
My question is the following
- Do there exist criterion on $p,X,S...$ to ensure the existence, at least (Zariski) locally, of a universal (or quasi-universal) family on $X\times M_{X/S}(P)$ ?
By quasi-universal, I mean that any family $\mathcal{E}$ on the fibres of $p$ parametrized by $S$-scheme $T$ is, up to the tensor by a vector bundle (not necessarily of rank $1$) on $S$, the pullback of the quasi-universal family by a $S$-map $T\to M_{X/S}(P)$.
When $S=Spec(\mathbb{C})$, there always exists a quasi-universal family (this is proved in Huybrechts-Lehn's book). On the other hand, according to Simpson here) there always exists etale locally a universal family.
Also, criterions exist when $M_{X/S}(P)$ is the Picard scheme $Pic_{X/S}$, the existence of a section $S\to X$ is such one, which is not easy to construct in general.
Any references treating a general or specific case is welcome. For instance, can we say something if we know that for all fibres $X_s$ the moduli space $M_{X_s}(P)$ is fine ?
