3 added 2 characters in body

Let $\overline{M}_{0,k}(\mathbb{P}^n,d)$

denote the moduli space of genus zero degree $d$ stable maps with $k$ marked points. This is an orbifold of expected dimension. Let $\overline{U}_{0,k}(\mathbb{P}^n,d)$ be the corresponding universal curve. Then we have two morphisms

$$\pi_1: \overline{U}{0,k}(\mathbb{P}^n,d) overline{U}_{0,k}(\mathbb{P}^n,d) \rightarrow \overline{M}{0,k}(\mathbb{P}^n,d)$$overline{M}_{0,k}(\mathbb{P}^n,d)$$ and$$\pi_2: \overline{U}_{0,k}(\mathbb{P}^n,d) \rightarrow \mathbb{P}^n$$So if we have a coherent sheaf F on \mathbb{P}^n we can pull it back to universal curve and push it forward to moduli space, i.e. we can consider F_i=(R^i\pi_{1*})(\pi_2^*F) on moduli space. Here are my questions: 1) Are F_i zero for i>0 ? (is \pi_{1*} exact ) 2) If F is equal to ideal sheaf of some smooth projective subvariety X, and \beta \in H_2(X) is of degree d, then is there in general any relation between some component of support of F_0 and the moduli space \overline{M}_{0,k}(X,\beta) ? 3) if the answer to question one is No, what is the interpretation of F_i, i>0, for the case when F is the ideal sheaf of X as in question two? 2 deleted 3 characters in body; added 4 characters in body Let \overline{M}_{0,k}(\mathbb{P}^n,d) denote the moduli space of genus zero degree d stable maps with k marked points. This is an orbifold of expected dimension. Let \overline{U}_{0,k}(\mathbb{P}^n,d) be the corresponding universal curve. Then we have two morphisms$$\pi_1: \overline{U}{0,k}(\mathbb{P}^n,d) \rightarrow \overline{M}{0,k}(\mathbb{P}^n,d)$$and$$\pi_2: \overline{U}_{0,k}(\mathbb{P}^n,d) \rightarrow \mathbb{P}^n$$So if we have a coherent sheaf F on \mathbb{P}^n we can pull it back to universal curve and push it forward to moduli space, i.e. we can consider F_i=(R^i\pi_{1*})(\pi_2^*F) on moduli space. Here are my questions: 1) Are F_i zero for i>0 ? (is \pi_{1*} exact ) 2) If F is equal to ideal sheaf of some smooth projective subvariety X, and \beta \in H_2(X) is of degree d, then is there in general any relation between some component of support of F_0 and the moduli space \overline{M}_{0,k}(X,\beta) ? 3) if the answer to question one is No, what is the interpretation of F_i, i>0, for the case when F is the ideal sheaf of X as in question two? 1 # Some questions on moduli of stable maps Let \overline{M}_{0,k}(\mathbb{P}^n,d) denote the moduli space of genus zero degree d stable maps with k marked points. This is an orbifold of expected dimension. Let \overline{U}_{0,k}(\mathbb{P}^n,d) be the corresponding universal curve. Then we have two morphisms$$\pi_1: \overline{U}{0,k}(\mathbb{P}^n,d) \rightarrow \overline{M}{0,k}(\mathbb{P}^n,d)$$and$$\pi_2: \overline{U}_{0,k}(\mathbb{P}^n,d) \rightarrow \mathbb{P}^n

So if we have a coherent sheaf $F$ on $\mathbb{P}^n$ we can pull it back to universal curve and push it forward to moduli space, i.e. we can consider $F_i=(R^i\pi_{1*})(\pi_2^*F)$ on moduli space.

Here are my questions:

1) Are $F_i$ zero for $i>0$ ? (is $\pi_{1*}$ exact )

2) If $F$ is equal to ideal sheaf of some smooth projective subvariety $X$, and $\beta \in H_2(X)$ is of degree $d$, then is there in general any relation between some component of support of $F_0$ and the moduli space $\overline{M}_{0,k}(X,\beta)$ ?

3) if the answer to question one is No, what is the interpretation of $F_i$, $i>0$, for the case when $F$ is the ideal sheaf of $X$ as in question two?