Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X. This is clearly a birationally invariant property. When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an… CONTINUE READING

A note on Hurwitz schemes of covers of a positive genus curve , preprint alggeom / 0205056 . [ H ] A . Hurwitz , Ueber Riemann ’ sche Flächen mit gegebenen Verzweigungspunkten ,