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I'm reading these notes

where it states in section $3$: (transcribed because I can't post image)

Step 1. Introduce the stacks of degenerated and iterated shtukas which extends that of shtukas.

This step is based on the well-studied scheme of complete homomorphisms of rank $r$ which is obtained from the scheme of non-zero $n \times n$ matrices by a series of blow-ups. Roughly speaking, the last condition $E^σ \xrightarrow{\sim} E^{''}$ will be replaced by a complete homomorphism $E^{σ} ⇒ E^{''}$ .

Referring to the construction of some kind of compactification of the stack of Drinfeld's Shtukas. What is this "well-studied scheme of morphisms via blowups" and what are any references? I don't even know what to google for a reference to this.

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  • $\begingroup$ The variety of complete homomorphisms of rank $r$ is the variety obtained by taking the determinantal variety of $n*n$ matrices of rank less or equal to $r$ and blowing-up successively the sub-variety of rank $0$ matrices, then the strict transform of rank $1$ matrices, then the strict tranform of the strict transform of the matrices of rank $2$ etc... up to $r-1$. This variety is smooth and this process is called a wonderful resolution of singularities. $\endgroup$ – Libli Sep 3 at 8:33
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The paragraph is written concisely so that it might be confusing, but it just means that there is a quite natural compactification of (truncated) moduli of Drinfeld shtukas where it is the moduli space of a moduli problem which relaxes the requirement of $E^{\sigma}\xrightarrow{\sim}E''$ being an isomorphism to requiring it to be just a "complete homomorphism." See Laumon's 2002 ICM article Section 5 for a quick definition of complete homomorphism, and I believe the note you referred to gives a bit of taste of what complete homomorphisms are in a later section. This is first observed (I think) by Laurent Lafforgue.

I agree that it is a little bit hard to google this at the first place partly because there is no universal terminology and partly because most references are in French. In particular iterated shtukas are nowadays used for something completely different. It will be safer to just look at Laurent Lafforgue's early papers. Maybe Une compactification des champs classifiant les chtoucas de Drinfeld is a good place to study this.

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