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 $X$?


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

Edit. Since user3001 mentioned my thesis, here is a link to my thesis.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.