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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

$$\pi_1: \overline{U}_{0,k}(\mathbb{P}^n,d) \rightarrow \overline{M}_{0,k}(\mathbb{P}^n,d)$$


$$\pi_2: \overline{U}_{0,k}(\mathbb{P}^n,d) \rightarrow \mathbb{P}^n$$

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?

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I am not sure if on your PC, you can see the Latex part correctly. Whatever I do, it does not show the line defining $\pi_1$ correctly. $\pi_1$ is the obvious map from universal curve to moduli space. –  Mohammad F. Tehrani Jun 23 '11 at 16:20
Motivation: This technique was used by some people to construct a virtual cycle for moduli spaces corresponding to quintic in $\mathbb{P}^4$ and using that plus localization they computed the GW invariants of quintic. There $F_0$ is a vector bundle on $\overline{M}(\mathbb{P}^4,d)$ and the zero locus of some section of that is a virtual fundamental cycle for degree $d$ maps to quintic. –  Mohammad F. Tehrani Jun 23 '11 at 16:55
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1 Answer

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

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You did not answer my 3rd question. What is the geometric meaning of higher derived sheaves? If we start with a vector bundle whose zero section is $X$ then what you say is true, but I'd like to start from the ideal sheaf of $X$ itself and then look at the support of induced sheaf on moduli space. In fact I want to know if this gives the same picture as in the case you mentioned. –  Mohammad F. Tehrani Jun 23 '11 at 19:03
Does anyone know a good reference for the fact in 2) that the virtual fundamental class of $\overline{M}_{0,k}(X,\beta)$ is the zero locus of that section? –  Dennis Ochse Sep 21 '12 at 8:35
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