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

Proof. 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. QED

Propoisiton 5. The natural transformation $\theta$ is relatively representable by proper schemes. Thus, the functor $F$ is representable.

Proof. Fix an integer $d$, and define $F^d$ to be the inverse image of $\text{Pic}^d_{C/k}$ with respect to $\theta$. It suffices to prove, for every $d\in \mathbb{Z}$, that the result holds for $$\theta^d:F^d \to \text{Pic}^d_{C/k}.$$ Fix a family of normalized invertible sheaves $(\mathcal{L},\tau)$ of relative degree $d$ on $C$ parameterized by $T$.

There exists an invertible sheaf $\mathcal{A}$ on $C$ such that for every invertible sheaf $\mathcal{B}$ of degree $\leq d$, for every invertible sheaf $\mathcal{B}$ of degree $\leq d$, resp. for every rank $1$, torsion coherent sheaf $\mathcal{T}$ of length $\leq d$, for every surjective homomorphism of coherent sheaves, $$V\otimes_k\mathcal{O}_C \twoheadrightarrow \mathcal{B}, \text{ resp. } \ \ V\otimes_k\mathcal{O}_C \twoheadrightarrow \mathcal{T},$$ then also the associated homomorphism, $$H^0(C,V\otimes_k\mathcal{A}) \rightarrow H^0(C,\mathcal{B}\otimes_{\mathcal{O}_C}\mathcal{A}), \text{ resp. } \ \ H^0(C,V\otimes_k\mathcal{A}) \rightarrow H^0(C,\mathcal{T}\otimes_{\mathcal{O}_C}\mathcal{A})$$ is surjective. In fact this is true precisely if $\text{deg}(A)-d\geq 2g-1$.

With respect to such an invertible sheaf $\mathcal{A}$, and the fixed normalized invertible sheaf $(\mathcal{L},\tau)$, the family of filtered $V$-systems on $C$ is uniquely determined by the system of morphisms, $$\phi_{i,\mathcal{A}}:V\otimes_k H^0(C,\mathcal{A})\otimes_k \mathcal{O}_T \to \text{pr}_{T,*}(\mathcal{L}\otimes \text{pr}_C^*\mathcal{A})|_{W_i}.$$ Since $\text{deg}(\mathcal{A})-d$ is greater than $2g-2$, all of the relevant higher direct image sheaves vanish, so that this sequence of morphisms is equivalent to a usual family of (irredundant) complete collineations from $H^0(C,V\otimes_k \mathcal{A})\otimes_k \mathcal{O}_T$ to $\text{pr}_{T,*}(\mathcal{L}\otimes \text{pr}_C^*\mathcal{A})$. The functor of complete collineations is representable. So it suffices to prove relative representability of the natural transformation associated to each family of filtered $V$-systems with specified normalized invertible sheaf the associated family of complete collineations.

First of all, this natural transformation is a monomorphism since $((W_0,\phi_0),\dots,(W_r,\phi_r))$ is uniquely determined by $(\phi_{0,\mathcal{A}},\dots,\phi_{r,\mathcal{A}})$. Thus, it suffices to prove that the image is representable by a locally closed subscheme of the space of complete collineations.

Define $\mathcal{F}_{C,\mathcal{A}}$ to be the kernel of the natural surjective homomorphism, $$H^0(C,\mathcal{A})\otimes_k \mathcal{O}_C\to \mathcal{A}.$$ For $s=0$, resp. for each integer $s=1,\dots,r$, for a family of complete collineations $(\psi_0,\dots,\psi_{s-1},\psi_s)$ as above, resp. for such a family with $\psi_i=\phi_{i,\mathcal{A}}$ for all $i<s$ coming from a filtered $V$-linear systems $((W_0,\phi_0),\dots,(W_{s-1},\phi_{s-1}))$ defined up to level $s-1$, define $\psi'_0$ to be the composite homomorphism of coherent sheaves, $$ V\otimes_k \text{pr}_T^*\mathcal{F}_{C,\mathcal{A}}\otimes_{\mathcal{O}_C} \to V\otimes_k H^0(C,\mathcal{A})\otimes_k \mathcal{O}_{T\times C} \xrightarrow{\psi_0} $$ $$\text{pr}_T^*\text{pr}_{T,*}(\mathcal{L}\otimes_{\mathcal{O}_{T\times C}} \text{pr}_C^*\mathcal{A}) \to \mathcal{L}\otimes_{\mathcal{O}_{T\times C}} \text{pr}_C^*\mathcal{A},$$ respectively, define $\psi'_s$ to be the composite homomorphism of coherent sheaves, $$ V\otimes_k \text{pr}_T^*\mathcal{F}_{C,\mathcal{A}}\otimes_{\mathcal{O}_C} \to V\otimes_k H^0(C,\mathcal{A})\otimes_k \mathcal{O}_{T\times C} \xrightarrow{\psi_s} $$ $$\text{pr}_T^*\text{pr}_{T,*}(\mathcal{L}\otimes_{\mathcal{O}_{T\times C}} \text{pr}_C^*\mathcal{A}|_{W_{s-1}}) \to \mathcal{L}\otimes_{\mathcal{O}_{T\times C}} \text{pr}_C^*\mathcal{A}|_{W_s}.$$ The necessary and sufficient condition for $\psi_0$ to equal $\phi_{0,\mathcal{A}}$ for a unique $\phi_0$, resp. for $\psi_s$ to equals $\phi_{s,\mathcal{A}}$ for a unique $\phi_s$, is vanishing of $\psi'_0$, resp. vanishing of $\psi'_s$. Working on the individual strata of the flattening stratification of $W_s$, vanishing of $\psi'_s$ defines a closed subset of each stratum. To prove closedness of the (constructible) union of these closed subsets of the (locally finite) union of the flattening strata, it suffices to prove properness of $\theta^d$, and for this it suffices to prove the valuative crtierion of properness.

Thus, assume that $T$ is Spec of a DVR and assume that $\phi_{s,\eta}$ is defined over the generic point of $T$. For $s=0$, it is straightforward to prove that there exists a unique extension $\phi_0$ over all of $T$ whose restriction to the closed fiber is not (identically) zero. For $s\geq 1$, there exists a $T$-flat and finite, closed subscheme $W'_s$ of $T\times_{\text{Spec}\ k}C$ that contains the zero scheme of $\phi_{s-1}$. By restricting $\phi_{s-1}$ and $\phi_{s,\eta}$ to $W'_s$ and pushing forward to $T$, we are reduced to the valuative criterion of properness for the usual functor of complete collineations (note, separatedness follows from the fact that the natural transformation is a monomorphism). QED

• $W_0$ equals $T$,
• $W_{i+1}$ equals the zero scheme of $\phi_i$ for every $i=1,\dots,r-1$,
• the homomorphism $\phi_r$ is surjective.

Definition 3. A family of filtered $V$-systems 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 $t:\text{Spec}\ \kappa \to \text{Spec}\ k$. For families $((\mathcal{L},\tau),(W_0,\phi_0),\dots,(W_d,\phi_d))$ and $((\mathcal{L}',\tau'),(W'_0,\phi'_0),\dots,(W'_d,\phi'_d))$, such that $W_i$ equals $W'_i$ for every $i$ (for which $W_i$ or $W'_i$ is nonempty), an equivalence from the first family to the second is 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 $f$-pullback of the family is the $(f\times \text{Id}_C)$-pullback of filtered $V$-linear system on $T\times_{\text{Spec}\ k} C$.

There is a functor $F$ on $k$-schemes that associates to every $k$-scheme $T$ the set of equivalence classes of families of filtered $V$-systems on $C$ parameterized by $T$. There is a natural transformation from this functor to the relative Picard functor. It is straightforward to prove that $F$ is relatively representable over the relative Picard functor. Of course both functors decompose with respect to the relative degree $d$ of the family of invertible sheaves. For fixed integer $d$, there exists a sufficiently ample invertible sheaf $\mathcal{A}$ on $C$ such that for every family of filtered $V$-systems on $C$, the invertible sheaf $\mathcal{L}\otimes \text{pr}_C^*\mathcal{A}$ is $\text{pr}_T$-relatively globally generated, so that the family is determined from 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 the special case that $C$ equals $\mathbb{P}^1$ and $V$ is free of rank $n$, 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 $X_1$.

• $W_0$ equals $T$,
• $W_{i+1}$ equals the zero scheme of $\phi_i$ for every $i=1,\dots,r-1$,
• the homomorphism $\phi_r$ is surjective (possibly because $W_r$ is empty).

Definition 3. A family of filtered $V$-systems 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 equivalent 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 $f$-pullback of the family is the $(f\times \text{Id}_C)$-pullback of filtered $V$-linear system on $T\times_{\text{Spec}\ k} C$.

Definition 4. The functor of families of filtered $V$-systems on $C$ 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 $(\mathcal{L},\tau)$.

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

Proof. 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. QED

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 $X_1$.