Fixed-domain curve counts for blow-ups of projective space

We study the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and virtual (Gromov-Witten) counts are found to agree asymptotically in the Fano (and some $(-K)$-nef) examples, but not in general. For toric blow-ups, geometric counts are expressed in terms of integrals on products of Jacobians and symmetric products of the domain curves, and evaluated explicitly in genus 0 and in the case of ${\text{Bl}}_q({\mathbb{P}}^r)$. Virtual counts for ${\text{Bl}}_q({\mathbb{P}}^r)$ are also computed via the quantum cohomology ring.