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Author:
(1) CALLA TSCHANZ.
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Table of Links
Abstract and Intro
Background on tropical perspective
The expanded construction
GIT stability
Stack perspective
The canonical moduli stack
References
3. The expanded construction
Output of expanded construction. The expanded degeneration X[n] ! C[n] which we construct in this section has the following properties:
3.1 The blow-ups
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.
3.2 Embedding into product of projective bundles
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.
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This paper is available on arxiv under CC 4.0 license.
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