# inclusion of moduli spaces induced by morphism between certain universal families

In the recent paper The desingularization of the theta divisor of a cubic threefold as a moduli space, they embed a cubic threefold $$X$$ into a certain moduli space of stable sheaves. But I do not know how explicitly it works

Namely, they consider a (smooth complex) cubic threefold $$X$$ as the moduli space of the twisted ideal sheaves $$\mathcal{I}_{x}\otimes\mathcal{O}_X(H)$$ for the natural polarization $$\mathcal{O}_X(H):=\mathcal{O}_{\mathbb{P}}(1)|_X$$. Then there exists a short exact sequence $$0\rightarrow \mathcal{K}_x\rightarrow\mathcal{O}_X^{\oplus 4}\rightarrow\mathcal{I}_x(H)\rightarrow0 \quad(\star)$$ where $$\mathcal{K}_x$$ is proved to be stable. Set $$v:=ch(\cal{K}^x)$$, it is proved that the moduli space $$\overline{M_X(v)}$$ is a smooth variety and only contains stable sheaves (i.e. $$\overline{M_X(v)}=\overline{M^s_X(v)}$$).

Let $$p\colon X\times X\rightarrow X$$ be the projection to the first factor, then there is a short exact sequence (the family version of the short short exact sequence $$(\star)$$): $$0\rightarrow \mathcal{K}\rightarrow p^*\Omega_{\mathbb{P}}|_X(H)\rightarrow\mathcal{I}_{\Delta}(0,H)\rightarrow0$$ It is asserted that this exact sequence induces the closed embedding $$i\colon X\hookrightarrow\overline{M_X(v)}$$.

The above statements are cited from their paper in particular Lemma 7.3. I want to know why the closed embedding is well-defined.

The moduli space of stable sheaves is by definition a scheme that corepresents the functor of families of stable sheaves. In particular, for any family of stable sheaves there is a canonical morphism from the base of the family to the moduli space. Applying this to the family of sheaves $$\mathcal{K}$$ one obtains the morphism $$i$$.
The moduli space $$M:=\overline{M_v(X)}=\overline{M^s_v(X)}$$ is the coarse moduli space corepresenting the functor $$\mathfrak{M}\colon ({\bf Sch }/\mathbb{C})^{\bf op }\rightarrow{\bf Sets}$$ given by $$\mathfrak{M}(T)=\{\mathcal{F}\in{\bf Coh}(X\times T)\,|\,\mathcal{F}_t\text{ is } H\text{-stable on }X\text{ and }ch(\mathcal{F}_t)=v\text{ for each }t\in T\}/\cong$$ In particular, there exists a natural transformation $$\alpha\colon\mathfrak{M}\rightarrow\hom(-,M)$$. Hence the morphism $$i\colon X\rightarrow M$$ is defined as the image of the family $$\mathcal{K}\in \mathfrak{M}(X)$$ under $$\alpha_X$$. Moreover, one checks that $$i\colon X\rightarrow M,\quad x\mapsto [\mathcal{K}_x]$$ according to the naturality of $$\alpha$$.