Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variation of Geometric Invariant Theory Quotients   Geometric Invariant Theory gives a method for constructing quotients for
group actions on algebraic varieties which in many cases appear as moduli
spaces parametrizing isomorphism classes of geometric objects (vector bundles,
polarized varieties, etc.). The quotient depends on a choice of an ample
linearized line bundle. Two choices are equivalent if they give rise to
identical quotients. A priori, there are infinitely many choices since there
are infinitely many isomorphism classes of linearized ample line bundles. Hence
several fundamental questions naturally arise. Is the set of equivalence
classes, and hence the set of non-isomorphic quotients, finite? How does the
quotient vary under change of the equivalence class? In this paper we give
partial answers to these questions in the case of actions of reductive
algebraic groups on nonsingular projective algebraic varieties. We shall show
that among ample line bundles which give projective geometric quotients there
are only finitely many equivalence classes. These classes span certain convex
subsets (chambers) in a certain convex cone in Euclidean space; and when we
cross a wall separating one chamber from another, the corresponding quotient
undergoes a birational transformation which is similar to a Mori flip.

31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph