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 $X_1$ ?

  • $\begingroup$ Your stratification is a "conical stratification" in the language of MacPherson and Procesi, "Making conical compactifications into wonderful ones". The iterative blowing up that you describe is what MacPherson-Procesi call the "minimal wonderful blowup". The blowing up you describe does not admit a $1$-morphism to the stack $M$ of genus $0$ stable maps to $\mathbb{P}^1\times \mathbb{P}^n$ of bidegree $(1,d)$. There is a $1$-morphism from $M$ to $\mathbb{P}^N$, and perhaps this factors through the blowing up. $\endgroup$ Feb 4 '18 at 16:21
  • 1
    $\begingroup$ Are you thinking about the wonderful compactification of $SL_{n+1}$? If so it seems to me that this sequence of blow-ups is a different compactification. Furthermore, the morphism from $M$ to $\mathbb{P}^N$ contracts the boundary divisors to subvarities of smaller dimension than the $Z_i$. So I do not think that the morphism $M\rightarrow \mathbb{P}^N$ factors through $X_1$. $\endgroup$
    – J_Cole
    Feb 4 '18 at 16:48
  • $\begingroup$ No, I am not thinking about the wonderful compactification of $\text{SL}_{n+1}$. There is a method for "making conical compactifications wonderful" that applies, in particular, to the compactifications of semisimple groups of adjoint type studied by DeConcini and Procesi. However, the theory of MacPherson and Procesi applies to many other conical stratifications than compactifications of semisimple groups. $\endgroup$ Feb 4 '18 at 16:54
  • $\begingroup$ You are completely correct that the inverse image of $Z_d$ in $M$ has irreducible components of codimension $>1$. Thus, the morphism from $M$ to $\mathbb{P}^N$ does not factor through $X_d$, much less $X_1$. $\endgroup$ Feb 4 '18 at 17:22

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 $X_1$.

  • $\begingroup$ Thank you very much for the great answer. Looking at your construction I am wondering if $X_{2}$ (we do not blow-up the bigger stratum) is precisely the space of complete collineations. Would this make any sense? $\endgroup$
    – J_Cole
    Feb 4 '18 at 22:00
  • $\begingroup$ I do not see, for $d\geq 2$, I do not believe that the blowing up equals a space of complete collineations. If $d$ equals $1$, then the blowing up does equal the space of complete collineations. I added some further details about the defining equations of the representing scheme of $F^d$ inside the space of complete collineations of $V\otimes_k H^0(C,\mathcal{A})$. $\endgroup$ Feb 5 '18 at 14:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.