Rigid curves on \(\overline M_{0,n}\) and arithmetic breaks (Q2895540)

The authors of the article construct rigid curves on \(\overline{M}_{0,n}\) that intersect the interior \(M_{0,n}\). The main motivation is that by a Theorem of \textit{S. Keel} and \textit{J. McKernan} [``Contractible Extremal Rays on \(\overline{M}_{0,n}\)'', \url{arXiv:alg-geom/9607009}] every hypothetical counterexample to the \(F\)-conjecture is a curve as above. The main idea, used in finding such curves, is that if a curve is an irreducible component of the exceptional set of a birational morphism, then it is rigid. The authors consider birational morphisms given by hypergraph constructions, special types of forgetful morphisms. These morphisms however have usually higher dimensional exceptional components. It is shown in the article that for one special case, given by the Hesse configuration, one can find a one dimensional irreducible component, and hence a rigid curve. It is also shown that unfortunately the found rigid curve does not give counterexample to the \(F\)-conjecture. That is, the found rigid curve is decomposed into a sum of \(F\)-curves using reduction mod \(p\).NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].