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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?

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