Properties of schemes of morphisms and applications to blow-ups

TL;DR

This paper studies the properties of morphism schemes from a fixed projective scheme to other schemes, showing they preserve limits and immersions, and applies these results to analyze rational curves on blow-ups of projective spaces.

Contribution

It establishes functorial properties of morphism schemes and applies these to classify and compute dimensions of rational curves on blow-ups.

Findings

Morphism schemes preserve limits and immersions.

Partition of schemes parametrizing rational curves on blow-ups.

Dimension bounds for components intersecting exceptional divisors.

Abstract

Let $X$ be a fixed projective scheme which is flat over a base scheme $S$. The association taking a quasi-projective $S$-scheme $Y$ to the scheme parametrizing $S$-morphisms from $X$ to $Y$ is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide an upper bound for the dimension of the irreducible components intersecting the exceptional divisors properly.