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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(1)$$\mathcal{I}_{x}\otimes\mathcal{O}_X(H)$ for the natural polarization $\mathcal{O}_X(1):=\mathcal{O}_{\mathbb{P}}(1)|_X$$\mathcal{O}_X(H):=\mathcal{O}_{\mathbb{P}}(1)|_X$. Then there exists a short exact sequence $$0\rightarrow \mathcal{K}_x\rightarrow\mathcal{O}_X^{\oplus 4}\rightarrow\mathcal{I}_x(1)\rightarrow0 \quad(\star)$$$$0\rightarrow \mathcal{K}_x\rightarrow\mathcal{O}_X^{\oplus 4}\rightarrow\mathcal{I}_x(H)\rightarrow0 \quad(\star)$$ where $\mathcal{K}_x$ is proved to be a stable sheaf with chern class $v$. ItSet $v:=ch(\cal{K}^x)$, it is proved that the moduli space $\overline{M_X(v)}$ is a smooth variety and indeed contains only contains stable coherent 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, they say that thethen there is a short exact sequence (the family version of the short short exact sequence $(\star)$): $$0\rightarrow \mathcal{K}\rightarrow p^*\Omega_{\mathbb{P}}|_X\rightarrow\mathcal{I}_{\Delta}(0,1)\rightarrow0$$$$0\rightarrow \mathcal{K}\rightarrow p^*\Omega_{\mathbb{P}}|_X(H)\rightarrow\mathcal{I}_{\Delta}(0,H)\rightarrow0$$ inducesIt is asserted that this exact sequence induces the closed embedding $i\colon X\hookrightarrow\overline{M_X(v)}$.

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

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 $\mathcal{I}_{x}\otimes\mathcal{O}_X(1)$ for the natural polarization $\mathcal{O}_X(1):=\mathcal{O}_{\mathbb{P}}(1)|_X$. Then there exists a short exact sequence $$0\rightarrow \mathcal{K}_x\rightarrow\mathcal{O}_X^{\oplus 4}\rightarrow\mathcal{I}_x(1)\rightarrow0 \quad(\star)$$ where $\mathcal{K}_x$ is proved to be a stable sheaf with chern class $v$. It is proved that the moduli space $\overline{M_X(v)}$ is a smooth variety and indeed contains only stable coherent 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, they say that the short exact sequence (the family version of $(\star)$) $$0\rightarrow \mathcal{K}\rightarrow p^*\Omega_{\mathbb{P}}|_X\rightarrow\mathcal{I}_{\Delta}(0,1)\rightarrow0$$ induces the closed embedding $i\colon X\hookrightarrow\overline{M_X(v)}$.

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

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.

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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 $\mathcal{I}_{x}\otimes\mathcal{O}_X(1)$ for the natural polarization $\mathcal{O}_X(1):=\mathcal{O}_{\mathbb{P}}(1)|_X$. Then there exists a short exact sequence $$0\rightarrow \mathcal{K}_x\rightarrow\mathcal{O}_X^{\oplus 4}\rightarrow\mathcal{I}_x(1)\rightarrow0 \quad(\star)$$ where $\mathcal{K}_x$ is proved to be a stable sheaf with chern class $v$. It is proved that the moduli space $\overline{M_X(v)}$ is a smooth variety and indeed contains only stable coherent 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, they say that the short exact sequence (the family version of $(\star)$) $$0\rightarrow \mathcal{K}\rightarrow p^*\Omega_{\mathbb{P}}|_X\rightarrow\mathcal{I}_{\Delta}(0,1)\rightarrow0$$ induces the closed embedding $i\colon X\hookrightarrow\overline{M_X(v)}$.

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