Some questions on moduli of stable maps

Question

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?

Answer 1

1) It depends on $F$. Roughly speaking fibers of $\pi_1$ are the curves parametrized by $\overline{M}_{0,k}(P^n,d)$. So if you want $R^{>0}\pi_{1*}\pi_2^*F$ to vanish, you have to check that $H^{>0}$ of the restriction of $F$ to any such curve vanishes. Since all the curves are rational it suffices (but not necessary) to assume that $F$ is generated by global section.

2,3) It seems that you want something else. The usual approach is the following. Assume that $X$ is the zero locus of a regular section $s$ of a vector bundle $F$. Then $s$ gives a global section of $\pi_2^*F$, and hence of $\pi_{1*}\pi_2^*F$. The claim is that (the virtual fundamental class of) $\overline{M}_{0,k}(X,\beta)$ is the zero locus of this section of $\pi_{1*}\pi_2^*F$ on $\overline{M}_{0,k}(P^n,d)$.