Pointed Evaluation Fibers of Rational Curves on del Pezzo Manifolds

Let $X$ be a Picard-rank-one del Pezzo manifold of dimension $n\geq 4$ over an algebraically closed field of characteristic zero.

Okamura proved that the unpointed Kontsevich spaces $\overline{M}_{0,0}(X,d)$ are irreducible of the expected dimension for every $d\geq 1$.

We refine this result by studying pointed evaluation fibers.

First, we prove that for every $d\geq 1$, the one-pointed evaluation morphism $\overline{M}_{0,1}(X,d)\to X$ has geometrically irreducible generic fiber.

Second, in the very ample cases $H^n=3,4,5$, we prove that for every $d\geq 2$, the two-pointed evaluation morphism $\overline{M}_{0,2}(X,d)\to X\times X$ has geometrically irreducible generic fiber.