Spaces of non-degenerate maps between complex projective spaces

Abstract

We study the space ${{\,\mathrm{Hol}\,}}_d(\mathbb {CP}^m,\mathbb {CP}^n)$ of degree d algebraic maps $\mathbb {CP}^m \rightarrow \mathbb {CP}^n$, from the point of view of homological stability as discovered by Segal (Acta Math 143(1–2):39–72, 1979) and later explored by Mostovoy (Topol Appl 45(2):281–293, 2006), Cohen et al. (Acta Math 166:163–221, 1991), Farb and Wolfson (N Y J Math 22:801–821, 2015), and others. In particular, we calculate the $\mathbb {Q}$-cohomology ring explicitly in the case $m=1$, as computed by Kallel and Salvatore (Geom Topol 10:1579–1606, 2006), and stably for when $m>1$. In doing so, we expand a method, previously studied by Crawford (J Differ Geom 38:161–189, 1993), for analyzing spaces of maps $X \rightarrow \mathbb {CP}^n$ by introducing subvarieties of non-degenerate functions that approximate the desired cohomologies both integrally and rationally in different ways. We also prove, when $m=n$, that the orbit space ${{\,\mathrm{Rat}\,}}_d(\mathbb {CP}^m,\mathbb {CP}^m)/{{\,\mathrm{PGL}\,}}_{m+1}(\mathbb {C})$ under the action on the target is $\mathbb {Q}$-acyclic up through dimension $d-2$, partially generalizing a calculation of Milgram (Topology 36(5):1155–1192, 1997).