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I was interested in the question of figuring out the rational homotopy type of mapping spaces (regular or rational) between two algebraic varieties over $\mathbb{C}$. I encountered the following paper "Contractibility of the space of rational maps". The paper for the most part is beyond me, but there seems to be a theorem that if $Y$ is a connected affine scheme covered by opens of $\mathbb{A}^n$ for some $n$. The space $Map_{rat}(X,Y)$ is contractible. One question that I had and I wasn't able to find an explicit description in the paper. What is the topology on $Map_{rat}(X,Y)$? Is it the same topology induced by the compact open topology on the space of continuous maps?

On paragraph above section 0.5.7., it mentions that this might be true under milder condition that requires $Y$ to be only smooth and birational.

I was wondering whether there are further results in this direction. I was particularly interested in the space of rational maps from $\mathbb{A}^m$ to something like $Sym^{\infty}(\mathbb{P}^r)$, is there any way to figure out its homotopy type maybe by considering a specific CW complex structure on $Sym^{\infty}(\mathbb{P}^r)$ and then seeing how each part is glued to each other after taking the $Map_{rat}(\mathbb{A}^m,-)$.

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    $\begingroup$ Taking $X$ and $Y$ to be $\Bbb{CP}^1$, the homotopy groups of the space ${\rm{Rat}}_d$ of morphisms $\Bbb{CP}^1\rightarrow\Bbb{CP}^1$ of degree $d$ have been studied in the literature. $\endgroup$
    – KhashF
    Jul 29 '21 at 18:25
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In Gaitsgory's setup, $Rat(X,Y)$ is not really a topological space (its an algebraic prestack). If you want to think about it topologically, then it is probably best to think of its associated homotopy type.

By definition, this homotopy type obtained as a homotopy colimit over the category $fset^{op}$ of finite sets and surjections of the following functor to $Top$: $$I \mapsto \{ f \in X^I, ~ g:(X-f(I)) \to Y \text{ regular}\},$$ where a surjection $h : I \to J$ acts by $(f,g) \mapsto (f \circ h, g)$.

A slightly tricky thing to extract is how the space assigned to $I$ is topologized, since it is inherently infinite dimensional. The idea is that it is naturally a union of finite dimensional topological spaces, which can be identified as schemes, and then the whole space is topologized as a colimit of the topological spaces underlying those schemes. More concretely, consider the space of regular maps $X - f(I) \to Y$ that have poles of order at most $n$ and take the union as $n \to \infty$.

Note that this whole setup is not at all equivalent to the setting that KhashF mentioned. In his sense, the space of degree $d$ rational maps $f: \mathbb P^1 \to \mathbb P^n$ is the same as the space of degree $d$ algebraic maps $f: \mathbb P^1 \to \mathbb P^n$, which is far from contractible. (In fact Segal proved that as $d \to \infty$ the homology of this space converges to the space of continuous maps $\mathbb P^1 \to \mathbb P^n$, which has nontrivial homology).

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