Relative geometric invariant theory
This paper provides new proofs for certain results on relative geometric invariant theory (relative GIT). GIT explores the structure of the quotient of an algebraic variety \(X\) by an algebraic group \(G\) and dates back to Hilbert who shows that if \(X\) is complex affine and \(G\) is complex reductive linear, the ring of invariants \(\mathbb{C}[X]^G\) is finitely generated, then we can define the quotient of \(X\) by \(G\) as the affine variety \(X/\!\!/ G=Spec(\mathbb{C}[V]^G)\). Mumford extends these notions to a more general setting, when \(X\) is projective, posing the notion of stability to remove certain unstable orbits in the quotient, measured by the so-called Hilbert-Mumford criterion. And also introducing the notion of linealization of the action to a line bundle to find affine \(G\)-invariant subsets of \(X\) to glue together in order to get a quotient of \(X\) by \(G\). The relative setting of this is given by a reductive linear algebraic group \(G\), quasi-projective algebraic varieties \(X\), \(Y\) with actions of \(G\), and a \(G\)-equivariant projective morphism \(\pi:X\rightarrow Y\). The question is whether there exists, given a linearization on \(Y\), a linearization on \(X\) relating the semistable points in both GIT quotients inducing a projective morphism \(\overline{\pi}:X/\!\!/ G \rightarrow Y/\!\!/ G\). When \(X\) and \(Y\) are projective, \textit{Z. Reichstein} [Invent. Math. 96, No. 2, 349--383 (1989; Zbl 0675.14021)] shows a positive solution for this problem, and Theorem 3.3 provides a simpler proof of this fact, based on a study of the minimal weights appearing in the Hilbert-Mumford criterion to measure stability when moving on the cone of one-parameter subgroups. \textit{L. H. Halle} et al. [``Relative VGIT and an application to degenerations of Hilbert schemes'', Preprint, \url{arXiv:1909.03780}] study this problem when \(Y\) is affine. The above argument is adapted to give an alternative proof of their result (c.f. Theorem 4.1) studying variations of GIT quotients (i.e. how stable points vary when moving the linearization). \textit{A. Białynicki-Birula} [Transform. Groups 3, No. 4, 301--319 (1998; Zbl 0940.14034)] finds a proof of the finiteness of possible \(G\)-invariant subsets being possible quotients, meaning that they can act as the subset of semistable points for a certain linealization. The paper of the author [Proc. Am. Math. Soc. 131, No. 2, 359--362 (2003; Zbl 1008.14009)] gives a simpler proof using the aforementioned technics on weights, and this is extended to the relative setting in this article (c.f. Theorem 5.4). Section 6 lists several situation where the relative GIT setting can help to study the geometry of the problem, say, Kirwan blow-ups, relative moduli spaces, degenerations of HIlbert schemes and constructions of Hitchin maps for decorated bundles and coherent systems.
variation of quotients and moduli spaces
scientific article; zbMATH DE number 7480962
scientific article