# Reference requence: scheme of complete homomorphisms of rank $r$ via blowups

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.

• 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. – Libli Sep 3 at 8:33

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.