Linear systems on moduli spaces of stable maps

Question

I am studying the general theory of moduli spaces of stable maps, in particuar of the moduli spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$ of degree $d$ stable maps from a rational curve with $n$ marked points to $\mathbb{P}^r$.

I see that there are morphisms $f_I:\overline{M}_{0,n}(\mathbb{P}^r,d)\rightarrow \overline{M}_{0,n-|I|}(\mathbb{P}^r,d)$ forgetting the marked points indexed by subsets $I\subset\{1,...,n\}$, and a morphism $g:\overline{M}_{0,n}(\mathbb{P}^r,d)\rightarrow \overline{M}_{0,n}$ forgetting the data of the map.

Let $D_{f_I}$ and $D_{g}$ be the divisors on $\overline{M}_{0,n}(\mathbb{P}^r,d)$ associated to $f_I$ and $g$.

Are the morphisms $f_I$ and $g$ induced by the linear systems $H^0(\overline{M}_{0,n}(\mathbb{P}^r,d),D_{f_I})$ and $H^0(\overline{M}_{0,n}(\mathbb{P}^r,d),D_{g})$ or may it happen that they are induced by proper linear subsystems of $H^0(\overline{M}_{0,n}(\mathbb{P}^r,d),D_{f_I})$ and $H^0(\overline{M}_{0,n}(\mathbb{P}^r,d),D_{g})$ ?

Thank you very much in advance.

Answer 1

I am posting this as an answer because the comment thread is too long. A proper morphism of integral, Noetherian schemes, $f:X\to Y$, is a contraction if the sheaf homomorphism $f^\#:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism. Typically we only talk about this when $X$, or at least $Y$, is normal, because then we have Zariski's Main Theorem and Stein Factorization. In particular, if $X$ and $Y$ are normal, then $f$ is a contraction if and only if the stalk of $f^\#$ at the generic point $\eta$ of $Y$ is an isomorphism, i.e., if and only if the $\kappa(\eta)$-algebra homomorphism, $$f^\#_\eta:\kappa(\eta) \to H^0(X_\eta,\mathcal{O}_{X_\eta}),$$ is an isomorphism. That holds if the geometric generic fiber of $f$ is integral.

Let $\mathcal{E}$ be any locally free $\mathcal{O}_Y$-module. Then by the Projection Formula, the natural $\mathcal{O}_Y$-module homomorphism, $$\mathcal{E}\otimes_{\mathcal{O}_Y} f_*\mathcal{O}_X \to f_*(f^*\mathcal{E}),$$ is an isomorphism. Thus, if $f$ is a contraction, then the adjunction homomorphism, $$\mathcal{E} \to f_*(f^*\mathcal{E}),$$ is an isomorphism. Thus, the associated map of global sections, $$\Gamma(Y,\mathcal{E})\to \Gamma(Y,f_*(f^*\mathcal{E})),$$ is an isomorphism. By the definition of pushforward, this is the same as the pullback map, $$\Gamma(Y,\mathcal{E})\to \Gamma(X,f^*\mathcal{E}).$$ In particular, for every invertible $\mathcal{O}_Y$-module $\mathcal{L}$, the adjunction homomorphism, $$\Gamma(Y,\mathcal{L})\to \Gamma(X,f^*\mathcal{L}),$$ is an isomorphism. Thus, in the classical language, the pullback of the complete linear system of $\mathcal{L}$ equals the complete linear system of $f^*\mathcal{L}$.

Regarding the specific morphisms $f_I$ and $g$, the moduli spaces involved are all projective, integral, and normal. Normality holds because these are coarse moduli spaces of smooth stacks. For a stack $\mathcal{X}$ that is regular and a dominant $1$-morphism to an integral algebraic space, $p:\mathcal{X}\to X$, by the universal property of the normalization, the morphism $p$ factors uniquely through the normalization $X^{\text{nor}}\to X$. Thus, if $p$ is initial among $1$-morphisms to algebraic spaces, i.e., if $p$ is a coarse moduli space of $\mathcal{X}$, then the normalization $X^{\text{nor}}\to X$ is an isomorphism. Thus the coarse moduli space of $\mathcal{X}$ is normal. Of course projectivity and integrality require other arguments.

Finally, because $f_I$ and $g$ are dominant morphisms of projective, integral, normal schemes, to prove that they are contractions it suffices to prove that the geometric generic fibers are integral. For each type of morphism, this is explained in the comments.