Author
Berczi, G
Jackson, J
Kirwan, F
Journal title
Journal of Differential Geometry 2017 Conference
DOI
10.4310/SDG.2017.v22.n1.a2
Last updated
2024-03-16T14:45:52.98+00:00
Abstract
<p>Variation of Geometric Invariant Theory (VGIT) [DH98, Tha96] studies the structure of the dependence of a GIT quotient on the choice of linearisation. This structure, and the concomitant wall-crossing picture relating the different quotients when a reductive group acts linearly on a projective variety (with respect to an ample linearisation), has long been a hallmark of classical GIT, and has found many diverse applications. In this article we show that under the conditions of results given in [BDHK16b] for non-reductive linear algebraic group actions on projective varieties, this structure persists in non-reductive GIT.</p> <br/> <p>In § 1 we review the key points of the classical theory, when a reductive linear algebraic group G acts on a projective variety X. Mumford’s GIT associates to any linearisation L of this action with respect to an ample line bundle L a notion of a quotient X//LG (where X//LG is a projective variety), and the variation of GIT results of Thaddeus [Tha96] and Dolgachev &amp; Hu [DH98] describe the dependence of X//LG on the linearisation L. In § 2 we give a brief exposition of how some of the difficulties characteristic of non-reductive GIT may be solved for actions of linear algebraic groups with graded unipotent radicals as in [BDHK16b] and [BDHK16a]. The next two sections are dedicated to showing that these results allow us to recover a variation picture which is very similar to the classical one, in the case where ‘semistability coincides with stability for the unipotent radical’, in a sense that we will specify later. The major difference from usual VGIT is the presence, a priori, of an additional parameter: a choice of a suitable 1-parameter subgroup of the group in question.</p> <br/> <p>We then discuss what happens without the simplifying assumption that semistability coincides with stability for the unipotent radical. Essentially the same description can be made to work, with some slight modifications. Finally, we discuss some illustrative examples, and indicate some potential applications of our results.</p>
Symplectic ID
847419
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Publication type
Conference Paper
Publication date
13 Sep 2018
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