Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Projective completions of graded unipotent quotients   The aim of this paper is to show that classical geometric invariant theory
(GIT) has an effective analogue for linear actions of a non-reductive algebraic
group $H$ with graded unipotent radical on a projective scheme $X$. Here the
linear action of $H$ is required to extend to a semi-direct product $\hat{H} =
H \rtimes \mathbb{G}_m$ with a multiplicative one-parameter group which acts on
the Lie algebra of the unipotent radical $U$ of $H$ with all weights strictly
positive, and which centralises a Levi subgroup $R \cong H/U$ of $H$.
  We show that $X$ has an $H$-invariant open subscheme (the 'hat-stable locus')
which has a geometric quotient by the $H$-action. This geometric quotient has a
projective completion which is a categorical quotient (indeed, a good quotient)
by $\hat{H}$ of an open subscheme of a blow-up of the product of $X$ with the
affine line; with additional blow-ups a projective completion which is itself a
geometric quotient can be obtained. Furthermore the hat-stable locus of $X$ and
the corresponding open subsets of the blow-ups of the product of $X$ with the
affine line can be described effectively using Hilbert-Mumford-like criteria
combined with the explicit blow-up constructions.
  Applications include the construction of moduli spaces of sheaves and Higgs
bundles of fixed Harder--Narasimhan type over a fixed nonsingular projective
scheme, and of moduli spaces of unstable projective curves of fixed
singularity. More recently, cohomology theory for reductive GIT quotients were
extended by the first and fourth author to the non-reductive situation studied
in this paper, and this was used to prove the Green--Griffiths--Lang and
Kobayashi hyperbolicity conjectures for generic projective hypersurfaces with
polynomial bounds on their degree.

33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat