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Let $X\subset \mathbb P^n$ be a smooth projective variety over $\mathbb C$ which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational cuves of degree $e>1$. Is it possible (are there examples) for an open subset of $M_{0,0}(X,e)$ to pametrize $e$-fold covers of lines of $X$?

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Yes, this happens for Del Pezzo surfaces, already for $X=\mathbb{P}^1 \times \mathbb{P}^1$ embedded in $\mathbb{P}^3$ as a smooth quadric surface. If you want an example where $\text{Pic}(X)\cong \mathbb{Z}$, this happens for every smooth cubic hypersurface $X$ in $\mathbb{P}^4$.

More generally, this will happen for all Fano manifolds that have "pseudo-index" equal to $1$ or $2$, i.e., for all Fano manifolds that contain a rational curve in $X$ whose anticanonical degree equals $1$ or $2$. This is one reason that some papers in this area include a hypothesis that the pseudo-index is at least $3$. That is also the reason that the recent theorem of Riedl-Yang on Kontsevich spaces of Fano hypersurfaces of index $>2$ is the best possible result.

Kontsevich spaces of rational curves on Fano hypersurfaces
Eric Riedl, David Yang
http://arxiv.org/abs/1409.3802

Edit. Since user3001 mentioned my thesis, here is a link to my thesis.
http://www.math.stonybrook.edu/~jstarr/j8.pdf

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  • $\begingroup$ Thank you very much (for the second example, theorem 62 in your thesis). $\endgroup$
    – user3001
    Mar 19 '16 at 14:53
  • $\begingroup$ Yes, in my thesis I proved that the Kontsevich spaces of genus $0$ stable maps to a smooth cubic threefold have precisely two components (for $e>1$): the generic point of the first component parameterizes embedded smooth rational curves, and the generic point of the second component parameterizes $e$-fold covers of lines in the cubic threefold. A similar picture holds for sufficiently general smooth hypersurfaces of degree $n-1$ in $\mathbb{P}^n$ with $n\geq 4$. However, only for $n=4$ can the result hold for every smooth hypersurface of degree $n-1$. $\endgroup$ Mar 19 '16 at 14:57

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