# Intermediate moduli spaces of stable maps

In the following paper:

A. Mustata, M. A. Mustata, "Intermediate moduli spaces of stable maps", Invent. math. 167, 47–90 (2007)

the authors introduced a variation on moduli spaces of stable maps denoted by $\overline{M}_{0,0}(\mathbb{P}^r,d,k)$. There exist a natural morphism $\overline{M}_{0,0}(\mathbb{P}^r,d)\rightarrow\overline{M}_{0,0}(\mathbb{P}^r,d,k)$ wich basically contracts the tails mapping with degree less than or equal to $d−k$.

Does there exist a version $\overline{M}_{0,n}(X,d,k)$ of these moduli spaces taking into account also the data of $n$ ordered points on the source curves, and $X$ is an arbitrary projective homogeneous variety?

• This depends on what you mean. In the papers of Coskun, Harris, and myself, we wrote down big, basepoint free divisor classes on the coarse moduli space $\overline{M}_{0,n}(G/P,\beta)$ that realize, among others, the contraction discovered also independenty by Adam Parker and Andrei and Magdalena Anca Mustata. As made clear in a paper with Coskun and Harris, this is very related (ultimately derivable from) a divisor class on $\overline{M}_{0,n}$ first written by Kawamata in his work on subadjunction, and proved to be basepoint free by Keel-McKernan. – Jason Starr Feb 26 '18 at 10:52
• . . . However, the maximal contractions of the coarse moduli space are not coarse moduli spaces of Deligne-Mumford stacks that extend the usual Deligne-Mumford stack over the maximal open where the contraction is an isomorphism. So if you are looking for a moduli interpretation that gives a Deligne-Mumford stack, that does not exist (there is an Artin stack whose good moduli space is the contraction). – Jason Starr Feb 26 '18 at 10:54
• Thank you very much for your answer. Is the paper you are referring to "THE AMPLE CONE OF THE KONTSEVICH MODULI SPACE"? It seems that here you are working with $X = \mathbb{P}^r$. – SMB Feb 26 '18 at 17:13
• I believe we have at least 3 papers on this topic (it was some time ago, perhaps I am confused). – Jason Starr Feb 26 '18 at 18:03

I am just adding some details to my comments above. Let $k$ be a field. Let $G$ be a simply connected, semisimple algebraic $k$-group; for simplicity, assume that $G$ is split. Let $X$ be a projective $k$-scheme with a transitive, smooth action of $G$. For simplicity, assume that $X$ has a $k$-point with stabilizer $P$ a ("standard", i.e., smooth) parabolic subgroup of $G$. The group of curve classes on $X$ equals $$\text{CH}_1(X)=\text{Hom}_{\mathbb{Z}}(\text{Pic}(X),\mathbb{Z}) = \text{Hom}_{\mathbb{Z}}(\text{Hom}_{k-\text{Gp. Sch.}}(P,\mathbb{G}_m),\mathbb{Z}).$$ For every curve class $\beta\in \text{CH}_1(X)$, denote by $\overline{\mathcal{M}}_{0,n}(X,\beta)$ the $k$-stack parameterizing families of genus-$0$, $n$-pointed stable maps to $X$ with pushforward curve class $\beta$, $$(C,(q_1,\dots,q_n),u:C\to X).$$ This is a smooth, proper algebraic stack with finite diagonal and projective coarse moduli space $\overline{M}_{0,n}(X,\beta)$; it is a Deligne-Mumford stack if the characteristic of $k$ is strictly larger than the degree $\langle c_1(L),\beta \rangle$ for every primitive ample generator $L$ of $\text{Pic}(X)$.
For every geometrically reduced, Cartier divisor $D\subset X$ denote by $U_D\subset \overline{\mathcal{M}}_{0,n}(X,\beta)$ the open substack parameterizing stable maps such that $u^{-1}D$ contains no irreducible component of $C$ and such that the effective Cartier divisor on $C$, $$u^*D + \underline{q}_1+\dots+\underline{q}_n,$$ contains no singular point of $C$ and has multiplicity $\leq 2$ at every point of the support. The complement $B_D$ of $U_D$ has codimension $\leq 2$ at every point. Denote by $m\geq 0$ the intersection number $\langle [D],\beta \rangle$. Considering $u^*D$ as an unordered collection of $m$ points on $C$, the stabilization of $(C,(q_1,\dots,q_m),u^*D)$ gives a point of the quotient $\overline{M}_{0,n+m}/\mathfrak{S}_m$ of $\overline{M}_{0,n+m}$ by the action of the symmetric group $\mathfrak{S}_m$ permuting the last $m$ points. Altogether, the defines a $1$-morphism, $$\rho_D:U_D\to \overline{M}_{0,n+m}/\mathfrak{S}_m.$$
For every Cartier divisor $A$ on $\overline{M}_{0,n+m}/\mathfrak{S}_m$, the pullback $\rho_D^*A$ extends to all of $\overline{\mathcal{M}}_{0,n}(X,\beta)$ since the complement $B_D$ has codimension $2$, and since the stack is smooth. Also, for every element $g\in G(k)$, there is the translate divisor $g\cdot D\subset X$. For every stable map, there is a dense open subscheme of $G$ parameterizing those $g$ such that the stable map is contained in $U_{g\cdot D}$. Since $G$ is rational, all of the divisor classes $\rho_{g\cdot D}^*A$ are linearly equivalent. Assuming that $A$ itself moves in a basepoint free linear system $(A_t)$, then the pullbacks $\rho_{g\cdot D}^*A_t$ span a basepoint free linear system on $\overline{\mathcal{M}}_{0,n}(X,\beta)$. This linear system defines a contraction of the coarse moduli space, $$\phi_{|A|}:\overline{M}_{0,n}(X,\beta) \to Y_{|A|}.$$ Of course the contraction only depends on the numerical equivalence class of $A$ up to positive multiple, and in fact it only depends on the corresponding contraction, $$\psi_{|A|}:\overline{M}_{0,n+m}/\mathfrak{S}_m \to Z_{|A|}.$$ These contractions were studied in the articles of Coskun, Harris, and me.
There are many contractions of $\overline{M}_{0,n+m}/\mathfrak{S}_m$, and each of these defines a contraction of $\overline{\mathcal{M}}_{0,n}(X,\beta)$. It is absurd to try to "chart the geography" of all such contractions, but some contractions / basepoint free divisors have many applications. The divisor that has, so far, had the most applications in algebraic geometry is the divisor class identified by Kawamata in his work on the subadjunction formula. In later work, Keel and McKernan proved that this divisor class is basepoint free. This $\mathfrak{S}_{n+m}$-invariant divisor class on $\overline{M}_{0,n+m}$ gives a contraction $\psi$ that contracts all boundary divisor classes except $\Delta_{2,n+m-2}$ (since the target of the contraction has Picard rank $1$, this is enough to uniquely specify the divisor class up to positive multiples).
Already when $X$ equals $\mathbb{P}^r_k$, when $D$ equals a hyperplane, when $m\geq 2$, when $n$ equals $0$, and when $A$ is Kawamata's divisor class, the associated contraction of $\overline{M}_{0,0}(\mathbb{P}^r_k,\beta)$ does not equal the coarse moduli space of a Deligne-Mumford stack equal to the usual Deligne-Mumford stack $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^r_k,\beta)$ over the maximal open $V$ of $Y_{|A|}$ where $\phi_{|A|}$ is an isomorphism. So there is no "modular interpretation" that is a Deligne-Mumford stack. Of course there is an Artin stack whose good moduli space equals $Y_{|A|}$ and restricts to $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^r_k,\beta)$ over $V$; the Artin stack coming from (unparameterized) quasimaps.