On the Topology and Arithmetic of Spaces of Non-degenerate Maps between Complex Projective Spaces

This thesis sets out to develop a general method for inductively studying spaces of maps into complex projective space in terms of subspaces of (non-)degenerate functions and to exhibit unexpected phenomenon therein. For historical reasons, we describe this method using the quasiprojective variety of degree d algebraic morphisms (a.k.a. holomorphic maps) ℂℙ^m → ℂℙ^n. for m ≤ n, as a primary example. In Chapters 3 and 4, we compute the associated ℚ-cohomology ring explicitly in the case m=1 and stably for when m>1, exhibiting homological stability as shown by Segal, Mostovoy, Farb-Wolfson, and others, as well as unexpected phenomenon regarding a particular subspace of degenerate maps. We also prove, when m=n, that the orbit space under the PGL action on the target is ℚ-acyclic up through dimension d-2, partially generalizing a result of Milgram. In Chapter 5, using point counts and the Grothendieck-Lefschetz trace formula in étale cohomology, we conclude with a homological density conjecture regarding the subspace of non-degenerate functions ℂℙ^1 → ℂℙ^n.