Let $\overline{M}_{0,0}(X, \beta)$ denote the moduli space of genus zero stable maps into $X$ that represent the homology class $\beta \in H_2(M, \mathbb{Z}) $. What does it mean to say that $~f:\mathbb{P}^1 \longrightarrow X $ is a ``free morphsim''? Secondly, what does it mean to say that $f$ is a free morphism that is birational onto its image?

Here $X$ is a compact complex surface.

  • $\begingroup$ "Birational onto its image" means just that: $f: \mathbf P^1 \rightarrow f(\mathbf P^1)$ is birational. "Free" means that $f^*T_X$ is globally generated. See for example Chapter 9 of math.ens.fr/~debarre/M2.pdf $\endgroup$ – Schemer Sep 10 '15 at 15:23
  • $\begingroup$ @Relapsarian: Can you give an example of something that is not free? In the reference you gave, on page 131, it says that a curve with image $C$ is free if and only if $C.C >= 0$. Does that mean a morphism $f:\mathbb{P}^1 \longrightarrow X $ is free if and only if $f[\mathbb{P}^1]. f [\mathbb{P}^1] >= 0$? $\endgroup$ – Ritwik Sep 10 '15 at 15:42
  • $\begingroup$ To the second question: yes! To the first: by the criterion you state, you need to find a $\mathbf P^1$ with negative selfintersection in a surface. Do you know an example of such a thing? $\endgroup$ – Schemer Sep 11 '15 at 11:07

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