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Hypothesis. There exists an invertible sheaf $\mathcal{L}$ on $X$ such that the $\mathcal{L}|_C$ is isomorphic to $\omega_{C/k}$ and such that $H^{n-2}(X,\mathcal{L})$ vanishes.

Hypothesis. The restriction map $H^0(X,\mathcal{L})\to H^0(C,\mathcal{L}|_C)$ is surjective.

Next, assume that $\omega_C$ is isomorphic to $\mathcal{O}_{\mathbb{P}^4}(1)|_C$, i.e., $\mathcal{L}$ equals $\mathcal{O}_{\mathbb{P}^4}(1)|_X$. Then from the defining short exact sequence above, it follows that $\mathcal{E}$ is stable. The slope is $\mu=c_1(\mathcal{O}_{\mathbb{P}^4}(1))/2$. An invertible sheaf $\mathcal{O}_{\mathbb{P}^4}(e)$ has slope $\geq \mu$ if and only if $e\geq 1$. Since there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e-1)|_X$ to $\mathcal{I}$ if and only if $e-1 \leq -1$, i.e., if and only if $e\leq 0$, and similarly there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e)|_X$ to $\mathcal{O}_X$ if and only if $e\leq 0$, there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e)|_X$ to $\mathcal{E}$ if and only if $e\leq 0$. Therefore $\mathcal{E}$ is stable.

It remains to describe the curves $C$. For every line $R$ in $X$, for every $2$-plane $\Pi$ that contains $R$, the intersection $\Pi\cap X$ is a plane quintic $R\cup C$, where $C$ is a plane curve of degree $4$. Thus, $C$ is a complete intersection curve with $H^0(C,\mathcal{O}_C)$ equal to $k$. Also $\omega_{C/k}$ equals $\mathcal{O}_{\mathbb{P}^4}(1)|_C$ by adjunction. Thus, these residual curves satisfy the hypotheses above. Note that $H^0(X,\mathcal{E})$ equals $3$, i.e., there is a net of such curves associated to $E$. This is consistent: there is also a net of $2$-planes $\Pi$ containing the line $R$ in $\mathbb{P}^4$.

Hypothesis 1. There exists an invertible sheaf $\mathcal{L}$ on $X$ such that the $\mathcal{L}|_C$ is isomorphic to $\omega_{C/k}$ and such that $H^{n-2}(X,\mathcal{L})$ vanishes.

Hypothesis 1. The restriction map $H^0(X,\mathcal{L})\to H^0(C,\mathcal{L}|_C)$ is surjective.

Hypothesis 3. The invertible sheaf $\omega_C$ is isomorphic to $\mathcal{O}_{\mathbb{P}^4}(1)|_C$, i.e., $\mathcal{L}$ equals $\mathcal{O}_{\mathbb{P}^4}(1)|_X$. Combined with Hypothesis 2, this is equivalent to the hypothesis that the embedding of $C$ in its linear span in $\mathbb{P}^4$ is a canonical embedding.

Under these hypotheses, from the defining short exact sequence above, it follows that $\mathcal{E}$ is stable. The slope is $\mu=c_1(\mathcal{O}_{\mathbb{P}^4}(1))/2$. An invertible sheaf $\mathcal{O}_{\mathbb{P}^4}(e)$ has slope $\geq \mu$ if and only if $e\geq 1$. Since there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e-1)|_X$ to $\mathcal{I}$ if and only if $e-1 \leq -1$, i.e., if and only if $e\leq 0$, and similarly there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e)|_X$ to $\mathcal{O}_X$ if and only if $e\leq 0$, there is a nonzero morphism from $\mathcal{O}_{\mathbb{P}^4}(e)|_X$ to $\mathcal{E}$ if and only if $e\leq 0$. Therefore $\mathcal{E}$ is stable.

It remains to describe the relevant family of canonical curves $C$. For every line $R$ in $X$, for every $2$-plane $\Pi$ that contains $R$, the intersection $\Pi\cap X$ is a plane quintic $R\cup C$, where $C$ is a plane curve of degree $4$. Thus, $C$ is a complete intersection curve with $H^0(C,\mathcal{O}_C)$ equal to $k$. Also $\omega_{C/k}$ equals $\mathcal{O}_{\mathbb{P}^4}(1)|_C$ by adjunction. Thus, these residual curves satisfy the hypotheses above.

Note that $H^0(X,\mathcal{E})$ equals $3$, i.e., there is a net of such curves associated to $E$. This is consistent: there is also a net of $2$-planes $\Pi$ containing the line $R$ in $\mathbb{P}^4$.

Before writing this long exact sequence, note first the canonical isomorphism, $$ \textit{Ext}^1_{\mathcal{O}_X}(i_*\mathcal{O}_C,i_*\mathcal{O}_C) \cong i_*(i^*\mathcal{I})^\vee = i_* N_{C/X}, $$ where $i^*\mathcal{I}$ is the conormal sheaf of $C$ in $X$, and where $N_{C/X}$ denotes the dual locally free $\mathcal{O}_C$-module, i.e., the normal sheaf. For every closed subscheme of arbitrary codimension $c$, equal to the rank of $N_{C/X}$, the exterior product on the Ext algebra defines a morphism of differential graded $i_*\mathcal{O}_C$-algebras, $$ i_* \bigwedge^\bullet_{\mathcal{O}_C} N_{C/X}^\vee \to \textit{Ext}^\bullet_{\mathcal{O}_X}(i_*\mathcal{O}_C,i_*\mathcal{O}_C). $$ Working locally, and using the fact that $C$ is a local complete intersection scheme, this morphism is an isomorphism. Chasing exact sequences, $\textit{Ext}^r_{\mathcal{O}_X}(i_*\mathcal{O}_C,\mathcal{O}_X)$ is nonzero if and only if $r$ equals the codimension $c$, in which case it equals the pushforward of the determinant of $N_{C/X}$. Chasing long exact sequences of the short exact sequence relating $\mathcal{I}$ and $i_*\mathcal{O}_C$, we have $$ \textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{F}) \cong \mathcal{F}, \ \ \textit{Ext}^{c-1}_{\mathcal{O}_X}(\mathcal{I},\mathcal{F}) \cong i_*(\text{det}(N_{C/X})\otimes_{\mathcal{O}_C} i^*\mathcal{F}), $$ and all other sheaf Ext groups vanish. In the case of interest, where $c$ equals $2$, this gives $$ \textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \cong \mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}, \ \ \textit{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \cong i_*\mathcal{O}_C. $$ Now, the long exact sequence of the Grothendieck spectral sequence becomes $$ 0\to H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \to \text{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \to H^0(C,\mathcal{O}_C) \xrightarrow{\delta} H^2(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}). $$ By Serre duality, the last term is dual to $H^{n-2}(X,\mathcal{L})$. This vanishes by hypothesis. Thus, there exists an extension class that maps to the element $1$ in $H^0(C,\mathcal{O}_C)$, and the set of all such extension classes is a torsor under the $k$-vector space $H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k})$. Interpreting the Ext group as a Yoneda Ext group, each such extension class is equivalent to a short exact sequence, $$ 0 \to \mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k} \to \mathcal{E}' \to \mathcal{I} \to 0, $$ such that the induced map $$ i^*\mathcal{E}' \to i^*\mathcal{I} $$ is an isomorphism. This is equivalent to local freeness of $\mathcal{E}'$ on a Zariski open neighborhood of $C$. Since also $\mathcal{E}'$ is locally free on the complement of $C$, since also $\mathcal{I}$ is an invertible sheaf on the complement of $C$, these extension classes are precisely the ones for which $\mathcal{E}'$ is locally free of rank $2$. The dual locally free sheaf $\mathcal{E}$ has a natural global section $s$ coming from $\textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{O}_X) \cong \mathcal{O}_X$. The set of isomorphism classes of pairs $(\mathcal{E},s)$ is equivalent to the set of isomorphism classes of Yoneda extension classes above. QED

Before writing this long exact sequence, note first the canonical isomorphism, $$ \textit{Ext}^1_{\mathcal{O}_X}(i_*\mathcal{O}_C,i_*\mathcal{O}_C) \cong i_*(i^*\mathcal{I})^\vee = i_* N_{C/X}, $$ where $i^*\mathcal{I}$ is the conormal sheaf of $C$ in $X$, and where $N_{C/X}$ denotes the dual locally free $\mathcal{O}_C$-module, i.e., the normal sheaf. For every closed subscheme of arbitrary codimension $c$, equal to the rank of $N_{C/X}$, the exterior product on the Ext algebra defines a morphism of differential graded $i_*\mathcal{O}_C$-algebras, $$ i_* \bigwedge^\bullet_{\mathcal{O}_C} N_{C/X} \to \textit{Ext}^\bullet_{\mathcal{O}_X}(i_*\mathcal{O}_C,i_*\mathcal{O}_C). $$ Working locally, and using the fact that $C$ is a local complete intersection scheme, this morphism is an isomorphism. Chasing exact sequences, $\textit{Ext}^r_{\mathcal{O}_X}(i_*\mathcal{O}_C,\mathcal{O}_X)$ is nonzero if and only if $r$ equals the codimension $c$, in which case it equals the pushforward of the determinant of $N_{C/X}$. Chasing long exact sequences of the short exact sequence relating $\mathcal{I}$ and $i_*\mathcal{O}_C$, we have $$ \textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{F}) \cong \mathcal{F}, \ \ \textit{Ext}^{c-1}_{\mathcal{O}_X}(\mathcal{I},\mathcal{F}) \cong i_*(\text{det}(N_{C/X})\otimes_{\mathcal{O}_C} i^*\mathcal{F}), $$ and all other sheaf Ext groups vanish. In the case of interest, where $c$ equals $2$, this gives $$ \textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \cong \mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}, \ \ \textit{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \cong i_*\mathcal{O}_C. $$ Now, the long exact sequence of the Grothendieck spectral sequence becomes $$ 0\to H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \to \text{Ext}^1_{\mathcal{O}_X}(\mathcal{I},\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}) \to H^0(C,\mathcal{O}_C) \xrightarrow{\delta} H^2(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k}). $$ By Serre duality, the last term is dual to $H^{n-2}(X,\mathcal{L})$. This vanishes by hypothesis. Thus, there exists an extension class that maps to the element $1$ in $H^0(C,\mathcal{O}_C)$, and the set of all such extension classes is a torsor under the $k$-vector space $H^1(X,\mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k})$. Interpreting the Ext group as a Yoneda Ext group, each such extension class is equivalent to a short exact sequence, $$ 0 \to \mathcal{L}^\vee\otimes_{\mathcal{O}_X}\omega_{X/k} \to \mathcal{E}' \to \mathcal{I} \to 0, $$ such that the induced map $$ i^*\mathcal{E}' \to i^*\mathcal{I} $$ is an isomorphism. This is equivalent to local freeness of $\mathcal{E}'$ on a Zariski open neighborhood of $C$. Since also $\mathcal{E}'$ is locally free on the complement of $C$, since also $\mathcal{I}$ is an invertible sheaf on the complement of $C$, these extension classes are precisely the ones for which $\mathcal{E}'$ is locally free of rank $2$. The dual locally free sheaf $\mathcal{E}$ has a natural global section $s$ coming from $\textit{Hom}_{\mathcal{O}_X}(\mathcal{I},\mathcal{O}_X) \cong \mathcal{O}_X$. The set of isomorphism classes of pairs $(\mathcal{E},s)$ is equivalent to the set of isomorphism classes of Yoneda extension classes above. QED

The answer to the second question is also negative. Here is a quick review of the Hartshorne-Serre construction in the smooth case. Let $X$ be a smooth, connected, proper $k$-scheme of pure dimension $n\geq 2$. Let $C\subset X$ be a codimension $2$, connected, closed subscheme that is a local complete intersection scheme.

A general quintic threefold $X$ has precisely $2875$ lines, and thus $M$ would contain $2875$ isolated points. However, there are many smooth quintic threefolds $X$ whose Fano scheme of lines has positive dimension, e.g., this holds if $X$ has a conical tangent hyperplane (the intersection with $X$ is a cone over a smooth plane quintic curve). There exists such a quintic threefold with a conical point and with an isolated line. For such a qunitic threefold, the moduli space $M$ of stable sheaves with fixed numerical invariants equal to those of $\mathcal{E}$ has one isolated component and one connected component that is a smooth plane quintic curve.

The answer to the second question is also negative. Here is a quick review of the Hartshorne-Serre construction in the smooth case. Let $X$ be a smooth, connected, proper $k$-scheme of pure dimension $n\geq 2$. Let $C\subset X$ be a codimension $2$, closed subscheme that is a local complete intersection scheme such that $H^0(C,\mathcal{O}_C)$ equals $k$ (this implies that $C$ is connected, and it is equivalent to connectedness if $C$ is reduced).

A general quintic threefold $X$ has precisely $2875$ lines, and thus $M$ would contain $2875$ isolated points. However, there are many smooth quintic threefolds $X$ whose Fano scheme of lines has positive dimension, e.g., this holds if $X$ has a conical tangent hyperplane (the intersection of the tangent hyperplane with $X$ is a cone over a smooth plane quintic curve). There exists such a quintic threefold with a conical point and with an isolated line. For such a qunitic threefold, the moduli space $M$ of stable sheaves with fixed numerical invariants equal to those of $\mathcal{E}$ has one isolated component and one connected component that is a smooth plane quintic curve.