Kontsevich spaces of rational curves on Fano hypersurfaces (Q668000)

This paper studies the Kontsevich moduli space $\overline{\mathcal M}_{0,0}(X,e)$ of stable maps of degree $e$ from rational curves to a general hypersurface $X$ of degree $d$ in $\mathbb P^n$. The main result shows that when $n\geq d+2$, the space $\overline{\mathcal M}_{0,0}(X,e)$ is an irreducible local complete intersection stack of expected dimension $e(n-d+1)+n-4$, which solves a conjecture of Coskun, Harris and Starr (except for the case $n = d+1$). To prove the result, the authors bound the codimension of the space of hypersurfaces for which the expected dimension of the moduli space does not hold, and use curves from nearby hypersurfaces to ensure that there are enough curves to apply Bend-and-Break.