First off, sorry if this seems vague.

Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa numbers notes is a category where:

  • the objects are $(R,S, \mu, \mathcal{P}, \gamma)$ where $R$ is a finitely generated $k$-algebra, $S$ a nonempty finite set, $\mu: S \rightarrow X(R)$ is a map of sets where $S$ is just a finite set, $\mathcal{P}$ is a $G$-bundle in the relative curve and $\gamma$ is a trivialization of of $\mathcal{P}$ over the open set $X_R \setminus |\mu(S)|$
  • The morphisms are compatible maps with $S \rightarrow S'$ being surjections (let's not spell it out here).

The idea behind this gadget is that it is the algebro-geometric analogue of the gadget $hocolim_{U_i} Maps_c(U_1, BG) \times ... \times Maps_c(U_i, BG)$ where the hocolim is taken over all disjoint open disks on $M$ a fixed manifold. Then nonabelian Poincare duality can be stated as a homology equivalence of the above prestack with $Bun_G(X)$.

My question: other than $G$-bundles have people thought about other kinds of algebro-geometric objects they wish to parametrize using the above approach?

Here's a possible example I have in mind which I have problems formulating: maybe we can take a look at a fixed scheme $Y \rightarrow X$ and I want to parametrize subschemes of $Y$. So its incarnation as a Ran space should be equivalent to marking certain points on the curve and above each of these points we have an irreducible closed subscheme of $Y$ and a "trivialization" (not sure what the appropriate notion here is) away from these points.

My question pertaining to this possible example is:

  • does it make sense?

  • what should trivialization mean in this context?

  • what should the "total space" of this Ran space be? In other words what is its analogue of $Bun_G$? Could it possibly be related to some type of Hilbert scheme of closed subschemes?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.