Pick a $k$-rational point $\sigma$ of $X$. Recall that $\Picardfunctor_{X/k}$ is isomorphic to the functor $\Picardfunctor_{X/k, \sigma}$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-chi-locally-constant-geometric} for every $d \in \mathbf{Z}$ there is an open subfunctor $$ \Picardfunctor^d_{X/k, \sigma} \subset \Picardfunctor_{X/k, \sigma} $$ whose value on a scheme $T$ over $k$ consists of those $\mathcal{L} \in \Picardfunctor_{X/k, \sigma}(T)$ such that $\chi(X_t, \mathcal{L}_t) = d + 1 - g$ and moreover we have $$ \Picardfunctor_{X/k, \sigma} = \coprod\nolimits_{d \in \mathbf{Z}} \Picardfunctor^d_{X/k, \sigma} $$ as fppf sheaves. It follows that the scheme $\underline{\Picardfunctor}_{X/k}$ (which exists by Proposition \ref{proposition-pic-curve}) has a corresponding decomposition $$ \underline{\Picardfunctor}_{X/k, \sigma} = \coprod\nolimits_{d \in \mathbf{Z}} \underline{\Picardfunctor}^d_{X/k, \sigma} $$ where the points of $\underline{\Picardfunctor}^d_{X/k, \sigma}$ correspond to isomorphism classes of invertible modules of degree $d$ on $X$. \medskip\noindent Fix $d \geq 0$. There is a morphism $$ \gamma_d : \underline{\Hilbfunctor}^d_{X/k} \longrightarrow \underline{\Picardfunctor}^d_{X/k} $$ coming from the invertible sheaf $\mathcal{O}(D_{univ})$ on $\underline{\Hilbfunctor}^d_{X/k} \times_k X$ (Remark \ref{remark-universal-object-hilb-d}) by the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}). Our proof of the representability of the Picard functor of $X/k$ in Proposition \ref{proposition-pic-curve} and Lemma \ref{lemma-open-representable} shows that $\gamma_g$ induces an open immersion on a nonempty open of $\underline{\Hilbfunctor}^g_{X/k}$. Moreover, the proof shows that the translates of this open by $k$-rational points of the group scheme $\underline{\Picardfunctor}_{X/k}$ define an open covering. Since $\underline{\Hilbfunctor}^g_{X/K}$ is smooth of dimension $g$ (Proposition \ref{proposition-hilb-d}) over $k$, we conclude that the group scheme $\underline{\Picardfunctor}_{X/k}$ is smooth of dimension $g$ over $k$. \medskip\noindent By Groupoids, Lemma \ref{groupoids-lemma-group-scheme-over-field-separated} we see that $\underline{\Picardfunctor}_{X/k}$ is separated. Hence, for every $d \geq 0$, the image of $\gamma_d$ is a proper variety over $k$ (Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-is-proper}). \medskip\noindent Let $d \geq g$. Then for any field extension $K/k$ and any invertible $\mathcal{O}_{X_K}$-module $\mathcal{L}$ of degree $d$, we see that $\chi(X_K, \mathcal{L}) = d + 1 - g > 0$. Hence $\mathcal{L}$ has a nonzero section and we conclude that $\mathcal{L} = \mathcal{O}_{X_K}(D)$ for some divisor $D \subset X_K$ of degree $d$. It follows that $\gamma_d$ is surjective. \medskip\noindent Combining the facts mentioned above we see that $\underline{\Picardfunctor}^d_{X/k}$ is proper for $d \geq g$. This finishes the proof of (2) because now we see that $\underline{\Picardfunctor}^d_{X/k}$ is proper for $d \geq g$ but then all $\underline{\Picardfunctor}^d_{X/k}$ are proper by translation. \medskip\noindent It remains to prove that $\gamma_d$ is smooth for $d \geq 2g - 1$. Consider an invertible $\mathcal{O}_X$-module $\mathcal{L}$ of degree $d$. Then the fibre of the point corresponding to $\mathcal{L}$ is $$ Z = \{D \subset X \mid \mathcal{O}_X(D) \cong \mathcal{L}\} \subset \underline{\Hilbfunctor}^d_{X/k} $$ with its natural scheme structure. Since any isomorphism $\mathcal{O}_X(D) \to \mathcal{L}$ is well defined up to multiplying by a nonzero scalar, we see that the canonical section $1 \in \mathcal{O}_X(D)$ is mapped to a section $s \in \Gamma(X, \mathcal{L})$ well defined up to multiplication by a nonzero scalar. In this way we obtain a morphism $$ Z \longrightarrow \text{Proj}(\text{Sym}(\Gamma(X, \mathcal{L})^*)) $$ (dual because of our conventions). This morphism is an isomorphism, because given an section of $\mathcal{L}$ we can take the associated effective Cartier divisor, in other words we can construct an inverse of the displayed morphism; we omit the precise formulation and proof. Since $\dim H^0(X, \mathcal{L}) = d + 1 - g$ for every $\mathcal{L}$ of degree $d \geq 2g - 1$ by Varieties, Lemma \ref{varieties-lemma-vanishing-degree-2g-and-1-line-bundle} we see that $\text{Proj}(\text{Sym}(\Gamma(X, \mathcal{L})^*)) \cong \mathbf{P}^{d - g}_k$. We conclude that $\dim(Z) = \dim(\mathbf{P}^{d - g}_k) = d - g$. We conclude that the fibres of the morphism $\gamma_d$ all have dimension equal to the difference of the dimensions of $\underline{\Hilbfunctor}^d_{X/k}$ and $\underline{\Picardfunctor}^d_{X/k}$. It follows that $\gamma_d$ is flat, see Algebra, Lemma \ref{algebra-lemma-CM-over-regular-flat}. As moreover the fibres are smooth, we conclude that $\gamma_d$ is smooth by Morphisms, Lemma \ref{morphisms-lemma-smooth-flat-smooth-fibres}.