Consider the projective space $\mathbb{P}^N$ parametrizing morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^n$, $f(x,y) = [f\_0(x,y):\dots:f\_n(x,y)]$ of degree $d$.


Let $Z\_i\subset\mathbb{P}^N$ be the locus of morphisms such that $f\_0,\dots,f\_n$ have $i$ common zeros, for $i = 1,\dots, d$. Then $Z\_d\subset Z\_{d-1}\subset \dots\subset Z\_{1}\subset\mathbb{P}^N$. Let $X\_h$ be the the variety obtained by blowing-up $Z\_d,\dots,Z\_h$ in order of increasing dimension. 


Is it known any modular interpretation of the varieties $X\_h$ and in particular of the last variety in this chain of blow-ups $X\_1$ ?



Blowing-up projective spaces of parametrized rational curves
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. 


**Definition 1.** For every $k$-scheme $T$, for every invertible $\mathcal{O}\_T$-module $\mathcal{L}$, a $(V,\mathcal{L})$-**system** 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 **zero scheme** 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 **filtered $V$-linear system** 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 * $W\_0$ 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).

 This is **irredundant** if $\phi\_i$ is nonzero whenever $W\_i$ is nonempty. For a $k$-morphism $f:T'\to T$, the **$f$-pullback** 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 $\mathbb{P}^1\_k$.


**Definition 2.** For every $k$-scheme $T$, a **family of normalized invertible sheaves** 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 **equivalence** 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 $d$.


**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 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**


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 $X\_1$.


