**Mutations of noncommutative crepant resolutions: Exchanges & Mutations of modifying modules**

by Eigenvector Initialization PublicationJune 11th, 2024

**Author:**

(1) CALLA TSCHANZ.

- Abstract and Intro
- Background on tropical perspective
- The expanded construction
- GIT stability
- Stack perspective
- The canonical moduli stack
- References

Output of expanded construction. The expanded degeneration X[n] ! C[n] which we construct in this section has the following properties:

In this expanded degeneration construction, we will be blowing up schemes along Weil divisors. A consequence of the way these blow-ups are defined is that the blow-up morphisms contract only components of codimension at least 2.

the morphisms corresponding to each individual blow-up. We therefore have the equality

We now fix the following terminology.

**Proposition 3.1.5.** *The following blow-up diagram commutes*

*Proof*. This is immediate from the local description of the blow-ups above.

We now extend the definition of ∆1-components to the schemes X[n] and fix some additional terminology.

Before we continue we fix some terminology which will help us describe the expanded components.

**Definition 3.1.11.** We refer to an irreducible component of a ∆-component as a bubble. The notions of two bubbles being equal and a bubble being expanded out in a certain fibre are as in Definitions 3.1.4 and 3.1.9.

Now, we note that there is a natural inclusion

which, in turn, induces a natural inclusion

*on the basis directions, and acts by*

*on the ∆-components.*

*Proof*. This follows immediately from [GHH19].

we described in the previous section are equivariant under the group action.

**Lemma 3.1.13.** *We have the isomorphism*

*Proof*. This is immediate from the above description of the group action.

*Remark* 3.1.14. We abuse notation slightly by referring to the group acting on X[n] by G, instead of G[n]. It should always be clear from the context what group G is meant.

**Lemma 3.2.1.** *There is an embedding*

From this, we deduce that there are embeddings

Hence we have embeddings

**Linearisations**. The following lemma gives a method to construct all the linearised line bundles we will need to vary the GIT stability condition.

This paper is available on arxiv under CC 4.0 license.

L O A D I N G

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