inclusion of moduli spaces induced by morphism between certain universal families

Question

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.

Answer 1

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 $i$.

Answer 2

Maybe let me explain Sasha's answer in more detail.

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 $\alpha$.