One modular interpretation of $X_1$ mimics the modular interpretation of the space of complete collineations.  Let $V$ be a $k$-vector space of finite dimension.  

<B>Definition 1.</B> For every $k$-scheme $T$, for every invertible $\mathcal{O}_T$-module $\mathcal{L}$, a $(V,\mathcal{L})$-<b>system</b> on $T$ is a homomorphism of $\mathcal{O}_T$-modules, $$\phi:V\otimes_k \mathcal{L}^\vee \to \mathcal{O}_T.$$  By adjointness of Hom and tensor, this is equivalent to a homomorphism $V\otimes_k \mathcal{O}_T \to \mathcal{L}.$  The <b>zero scheme</b> of $\phi$ is the closed subscheme of $T$ whose ideal sheaf equals the image of $\phi$.  This closed subscheme is empty if and only if $\phi$ is surjective, i.e., if and only if $(V,\mathcal{L},\phi)$ is a linear system in the usual sense.  A <b>filtered $V$-linear system</b> on $T$ is a datum $$(\mathcal{L},(W_0,\phi_0),\dots,(W_r,\phi_r))$$ of an invertible $\mathcal{O}_T$-module $\mathcal{L}$, and an ordered sequence of pairs $(W_i,\phi_i)$ of a closed subscheme $W_i\subset T$ and a $(V,\mathcal{L}|_{W_i})$-systems $\phi_i$ on $W_i$ such that <ul><li> $W_0$ equals $T$, <li> $W_{i+1}$ equals the zero scheme of $\phi_i$ for every $i=1,\dots,r-1$, <li> the homomorphism $\phi_r$ is surjective (possibly because $W_r$ is empty). </ul>  This is <b>irredundant</b> if $\phi_i$ is nonzero whenever $W_i$ is nonempty.  For a $k$-morphism $f:T'\to T$, the <b>$f$-pullback</b> equals the datum $$(f^*\mathcal{L},(T'\times_T W_0,f^*\phi_0),\dots,(T'\times_T W_r,f^*\phi_r)).$$ 

Now let $C$ be a smooth, projective, geometrically connected $k$-curve.  For simplicity, assume that there exists a $k$-point $$p:\text{Spec}\ k \to C.$$ This is true when $C$ equals $\mathbb{P}^1_k$.

<B>Definition 2.</B> For every $k$-scheme $T$, a <b>family of normalized invertible sheaves</B> on $C$ parameterized by $T$ is a pair $(\mathcal{L},\tau)$ of an invertible sheaf $\mathcal{L}$ on $T\times_{\text{Spec}\ k}C$ together with an isomorphism, $$\tau:(\text{Id}_T \times p)^*\mathcal{L} \xrightarrow{\cong} \mathcal{O}_T.$$  An <b>equivalence</b> of families of normalized invertible sheaves on $C$ parameterized by $T$, $$\sigma:(\mathcal{L},\tau)\to (\mathcal{L}',\tau'),$$ is an isomorphism of invertible sheaves, $$\sigma:\mathcal{L}\to \mathcal{L}',$$ such that $\tau'\circ (\text{Id}_T\times p)^*\sigma$ equals $\tau$.  

The functor of equivalence classes of normalized invertible sheaves is represented by a group $k$-scheme of $C$ over $\text{Spec}\ k$, $$\text{Pic}_{C/k} = \bigsqcup_{d\in \mathbb{Z}} \text{Pic}^d_{C/k},$$ where $\text{Pic}^0_{C/k}$ is an Abelian $k$-variety of dimension $g$ equal to the arithmetic genus of $C$, and where every connected component $\text{Pic}^d_{C/k}$ is a torsor for $\text{Pic}^0_{C/k}$ parameterizing families of normalized invertible sheaves of relative degree $d$.

<B>Definition 3.</B> A <b>family of filtered $V$-systems</b> on $C$ parameterized by $T$ is a datum $$ ((\mathcal{L},\tau),(W_0,\phi_0),\dots,(W_r,\phi_r)) $$ of a family $(\mathcal{L},\tau)$ of normalized invertible sheaves on $C$ parameterized by $T$ and a filtered $V$-linear system $(\mathcal{L},(W_0,\phi_0),\dots,(W_d,\phi_d))$ on $T\times_{\text{Spec}\ k}C$ having irredundant restriction to the fiber $\text{Spec}\ \kappa \times_{\text{Spec}\ k}C$ over every geometric point $t:\text{Spec}\ \kappa \to \text{Spec}\ k$.  Families $((\mathcal{L},\tau),(W_0,\phi_0),\dots,(W_d,\phi_d))$ and $((\mathcal{L}',\tau'),(W'_0,\phi'_0),\dots,(W'_e,\phi'_e))$, are <b>equivalent</b> if $W_i$ equals $W'_i$ whenever at least one of $W_i$ or $W'_i$ is nonempty, and there exists a sequence of isomorphisms of invertible sheaves, $$\sigma_i:\mathcal{L}|_{W_i}\to \mathcal{L}|_{W'_i},$$ that commute with $\phi_i$ and $\phi'_i$. For every $k$-morphism $f:T'\to T$, the <b>$f$-pullback</b> of the family is the $(f\times \text{Id}_C)$-pullback of filtered $V$-linear system on $T\times_{\text{Spec}\ k} C$.

<B>Definition 4.</B> The <b>functor of families of filtered $V$-systems on $C$</b> is the contravariant functor $F$ on $k$-schemes associating to every $k$-scheme $T$ the set of equivalence classes of families of filtered $V$-systems on $C$ parameterized by $T$ and associating to every $k$-morphism $f$ the $f$-pullback map on equivalence classes of families of filtered $V$-systems.  There is a natural transformation $\theta$ from $F$ to the relative Picard functor associated to each family of filtered $V$-systems the family of normalized invertible sheaves $(\mathcal{L},\tau)$.

<B>Propoisiton 5.</B>  The natural transformation $\theta$ is relatively representable.  Thus, the functor $F$ is representable.

<B>Proof.</B>  Fix an integer $d$, and define $F^d$ to be the inverse image of $\text{Pic}^d_{C/k}$ with respect to $\theta$.  There exists an invertible sheaf $\mathcal{A}$ on $C$ such that for every family of normalized invertible sheaves of relative degree $d$, the invertible sheaf $\mathcal{L}\otimes_{\mathcal{O}_{T\times C}}\text{pr}_C^*\mathcal{A}$ is $\text{pr}_T$-relatively globally generated.  In fact this is true precisely if $\text{deg}(A)+d \geq 2g$. 

With respect to such an invertible sheaf $\mathcal{A}$, and for a fixed normalized invertible sheaf $(\mathcal{L},\tau)$, the family of filtered $V$-systems on $C$ is uniquely determined by the system of morphisms, $$\phi_{i,\mathcal{A}}:V\otimes_k H^0(C,\mathcal{A})\otimes_k \mathcal{O}_T \to \text{pr}_{T,*}(\mathcal{L}\otimes \text{pr}_C^*\mathcal{A})|_{W_i}.$$  In particular, the filtered $V$-system is uniquely by the system of images, considered as quotient sheaves of the fixed sheaf $V\otimes_k H^0(C,\mathcal{A})\otimes_k \mathcal{O}_T$.  From this point we proceed as in the usual construction of the space of complete collineations: construct the representing scheme as the closure inside a fiber product of Grassmannians of $V\otimes_k H^0(C,\mathcal{A})$ of the locally closed subscheme that represents the open subfunctor of $F^d$ parameterizing families with $\phi_0$ surjective. <B>QED</B>

In the special case that $C$ equals $\mathbb{P}^1$ and $V$ is free of rank $n+1$, so that $\mathbb{P}V$ is $k$-isomorphic to $\mathbb{P}^n_k$, the component $F^d$ of $F$ over $\text{Pic}^d_{\mathbb{P}^1_k/k} = \text{Spec}\ k$ equals the blowing up $X_1$.