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Geometric invariant theory

Geometric invariant theory (GIT) is a branch of algebraic geometry that provides a framework for constructing quotients of algebraic varieties under the action of reductive algebraic groups, using geometric invariants to separate orbits and form moduli spaces.^[1] It addresses the problem of when an orbit space exists as an algebraic variety, focusing on projective varieties and linearizations of group actions to ensure well-behaved quotients.^[2] Developed by David Mumford in his seminal 1965 monograph, GIT builds on classical invariant theory, particularly Hilbert's finiteness theorem, which guarantees that the ring of invariants is finitely generated when the acting group is reductive (such as GL(n) or finite groups).^[1] The theory was expanded in subsequent editions, including contributions from John Fogarty and Frances Kirwan, with the third edition in 1994 adding topics like the moment map for symplectic actions.^[1] Central to GIT is the notion of stability: points are semi-stable if their orbit closures (in the affine cone) do not contain the origin under any one-parameter subgroup action, and stable points form an open subset where orbits are closed with finite stabilizers. This is formalized by the Hilbert-Mumford numerical criterion, which characterizes stable and semi-stable points to define the quotient.^[2] GIT quotients, denoted X//G, are Proj of the invariant sections of ample line bundles, yielding projective varieties that parameterize orbits of semi-stable points and serve as coarse moduli spaces.^[2] This construction is pivotal in algebraic geometry for studying moduli spaces of curves, vector bundles, and sheaves, bridging geometric and categorical approaches to classification problems.^[1] Applications extend to symplectic geometry via Kirwan's work and to stacky quotients, where GIT provides approximations to Deligne-Mumford stacks.^[3]

Historical Development

Classical Invariant Theory

Classical invariant theory emerged in the 19th century as the algebraic study of polynomial functions that remain unchanged under linear transformations, primarily focusing on actions of groups like GL(n) or SL(n) on vector spaces of forms. Arthur Cayley and James Joseph Sylvester were pivotal figures in this development, with Cayley introducing the concept of covariants—polynomials that transform by a scalar factor under group actions—and Sylvester advancing the classification of invariants and covariants for binary forms through symbolic methods. Their work emphasized explicit computations, producing tables of generating invariants for binary forms up to degree 10, though Cayley initially erred in conjecturing that no finite basis exists for binary forms of degree 7 or higher. Syzygies, the algebraic relations among these invariants and covariants, were a key focus; for instance, for binary cubics, the syzygy T^2 = 2^4 3^6 A Q^2 - H^3 relates the covariants T, A, Q, and H.^[4] A cornerstone example is the action of SL(2) on binary forms, homogeneous polynomials f(x, y) = \sum_{i=0}^d a_i x^{d-i} y^i of degree d, where the group acts via linear substitutions (x, y) \mapsto (a x + b y, c x + d y) with ad - bc = 1. The discriminant serves as a fundamental invariant: for a binary quadratic, it is D_2 = a_0 a_2 - a_1^2 of degree 2, vanishing when the form has a double root; for a binary cubic, D_3 = 18 a_0 a_1 a_2 a_3 - 4 a_1^3 a_3 + a_1^2 a_2^2 - 4 a_0 a_2^3 - 27 a_0^2 a_3^2 (up to scalar) of degree 4 and index 6, similarly detecting multiple roots. For ternary cubics, forms in three variables of degree 3 under SL(3), the invariant ring is generated by polynomials S of degree 4 (with 25 monomials) and T of degree 6 (with 103 monomials), often computed via Lie algebra methods or the σ-process.^[5]^[6] David Hilbert's 1890 finiteness theorem marked a turning point, proving that for a linearly reductive group G (such as SL(n), GL(n), or finite groups) acting linearly on a finite-dimensional complex vector space V, the ring of invariants \mathbb{C}[V]^G is finitely generated as a \mathbb{C}-algebra. This generalized Gordan's 1868 result for SL(2) on binary forms, where Gordan's proof directly refuted Cayley's conjecture for binary forms, and resolved the existence of finite bases without relying on constructive algorithms, using the Reynolds operator and Hilbert's basis theorem for ideals. The proof applies to reductive groups, ensuring complete reducibility of representations, and holds over algebraically closed fields of characteristic zero.^[7] Despite these advances, classical invariant theory faced significant challenges, particularly in computational complexity for higher dimensions and forms of greater degree, where explicit generation of invariants becomes infeasible due to exponential growth in monomials—for example, the invariant T for ternary cubics initially expands to over 18,000 terms before simplification. Methods like symbolic computation or Gröbner bases, while effective for binary forms up to degree 8, scale poorly for m-ary forms with m > 2, limiting practical applications. Moreover, the quotients formed by invariant rings often lacked geometric interpretations, treating orbits algebraically without addressing stability or projective structures, which hindered broader applications until geometric methods arose in the 20th century.^[5]^[8]

Mumford's Geometric Approach

David Mumford introduced a geometric reformulation of invariant theory in his 1965 monograph Geometric Invariant Theory, shifting the focus from classical algebraic methods to tools from algebraic geometry. This work addressed limitations exposed by Masayoshi Nagata's 1959 counterexamples to Hilbert's fourteenth problem, which showed that rings of invariants under linear group actions are not always finitely generated, complicating quotient constructions in the affine setting.^[9] Mumford's approach provided a framework for building moduli spaces even in such cases, emphasizing reductive group actions on projective varieties to yield well-behaved geometric quotients.^[10] A key innovation was the transition from affine varieties to projective ones, where the affine space is embedded into projective space via the sections of an ample line bundle, enabling a compatible linearization of the group action on the bundle. In the affine context, quotients often fail to be separated or geometric due to infinite stabilizers or non-finite generation of invariants, leading to pathologies like non-Hausdorff orbit spaces.^[11] Projective embeddings resolve these by restricting to finite-dimensional subspaces of invariant sections, ensuring the resulting quotient is projective and captures the orbit structure more robustly.^[12] The core idea centers on constructing geometric quotients via orbits under reductive group actions, where stability conditions select an open subset of points with favorable stabilizer and closure properties, allowing the quotient to inherit desirable geometric features like properness. Mumford's book structures this theory around the study of algebraic group actions on varieties, orbit separation using invariants, and explicit quotient constructions via the Proj functor applied to graded rings of invariant sections from linearized ample line bundles, with applications to moduli problems.^[1] This framework not only revives Hilbert's finiteness ideals in a geometric guise but also extends invariant theory to broader algebraic-geometric contexts.^[13]

Fundamental Setup

Reductive Group Actions on Projective Varieties

A reductive algebraic group G over an algebraically closed field k is defined as a smooth connected linear algebraic group whose unipotent radical R_u(G), the maximal connected normal unipotent subgroup, is trivial, i.e., R_u(G) = \{e\}.^[14] This condition ensures that G lacks nontrivial unipotent normal subgroups, allowing for representations that are completely reducible over k.^[14] Prominent examples include the general linear group \mathrm{GL}(n, k), the special linear group \mathrm{SL}(n, k), and the projective general linear group \mathrm{PGL}(n, k), all of which arise naturally in linear algebra and play central roles in constructing invariants.^[14] An action of such a group G on an algebraic variety X over k is specified by a morphism \alpha: G \times X \to X satisfying \alpha(e, x) = x and \alpha(g, \alpha(h, x)) = \alpha(gh, x) for all g, h \in G and x \in X, typically denoted by g \cdot x = \alpha(g, x).^[15] For affine varieties, the action induces a contravariant action on the coordinate ring \mathcal{O}(X) via (g \cdot f)(x) = f(g^{-1} \cdot x) for f \in \mathcal{O}(X).^[15] On projective spaces \mathbb{P}^n, actions are induced by linear representations of G on the underlying vector space k^{n+1}, acting on homogeneous coordinates and preserving the projectivization.^[15] In geometric invariant theory, the variety X is taken to be projective to ensure compactness in the classical topology, which is essential for forming quotients that are themselves projective varieties and suitable for moduli problems.^[16] Projectivization compactifies affine actions, preventing orbits from escaping to infinity and enabling the Proj construction of invariant rings to yield geometrically meaningful quotients.^[16] For a point x \in X, the orbit G \cdot x = \{g \cdot x \mid g \in G\} is a constructible subset of X, and its Zariski closure \overline{G \cdot x} may strictly contain the orbit itself.^[15] An orbit is closed if it equals its closure. The stabilizer \mathrm{Stab}_G(x) = \{g \in G \mid g \cdot x = x\} forms a closed subgroup of G, and the dimension of the orbit satisfies \dim(G \cdot x) = \dim G - \dim \mathrm{Stab}_G(x).^[15] Mumford's geometric approach to invariant theory leverages these orbit structures to interpret classical invariants geometrically.^[15]

Linearizations of Line Bundles

In the framework of reductive group actions on projective varieties, a linearization equips an ample line bundle with a compatible group action, allowing the study of weighted invariants through global sections. A linearization of a line bundle L on a scheme X with respect to an action of an algebraic group G on X is an algebraic action of G on the total space of L such that the projection map L \to X is G-equivariant. Equivalently, it consists of a collection of isomorphisms \phi_g: g^* L \to L for each g \in G, satisfying the cocycle condition \phi_{gh} = \phi_g \circ (g^* \phi_h) and such that each \phi_g is linear on the fibers. For reductive groups, such linearizations often involve a character \chi: G \to \mathbb{G}_m, twisting the isomorphism as g^* L \cong L \otimes \chi(g), which ensures compatibility with the group structure and enables the action to descend to projective quotients. This structure is foundational in geometric invariant theory, as introduced by Mumford. Ample line bundles play a central role in this setup by providing embeddings of the variety into projective space. Specifically, if L is ample on the projective variety X, then for sufficiently large r, the bundle L^{\otimes r} is very ample, meaning the map X \to \mathbb{P}(H^0(X, L^{\otimes r})^*) given by the complete linear system |L^{\otimes r}| is a closed embedding. A linearization on such an L lifts the G-action to this projective embedding, ensuring that the action on sections H^0(X, L^{\otimes r}) is algebraic and preserves the ample cone, which is crucial for finite generation of invariant rings. This embedding property allows GIT to translate abstract actions into concrete computations in projective space. The invariant sections under a linearized action form the algebraic backbone of GIT. For a linearized ample line bundle L on X, the G-invariant global sections H^0(X, L^{\otimes r})^G generate, for large r, a finitely generated graded algebra R(X, L)^G = \bigoplus_{r \geq 0} H^0(X, L^{\otimes r})^G. These sections serve as the basis for the ring of invariants, enabling the Proj construction of the categorical quotient while respecting the group action's geometry. The choice of linearization influences the grading and thus the resulting invariants, with ample L guaranteeing that the algebra is generated in sufficiently low degrees. A canonical example arises from the action of \mathrm{SL}(n) on \mathbb{P}^{n-1}, the projectivization of \mathbb{C}^n. Here, \mathrm{SL}(n) acts by matrix multiplication: g \cdot = [g v] for g \in \mathrm{SL}(n) and \in \mathbb{P}^{n-1}. The hyperplane bundle \mathcal{O}(1) on \mathbb{P}^{n-1} admits a natural linearization induced by this action, as \mathrm{SL}(n) preserves the determinant (character \chi = 1), lifting the action to the total space of \mathcal{O}(1) fiberwise linearly without twist. The invariant sections H^0(\mathbb{P}^{n-1}, \mathcal{O}(r))^{\mathrm{SL}(n)} are trivial for r > 0 due to the irreducibility of the representation, yielding a GIT quotient that is a point, illustrating how linearizations capture the absence of nontrivial invariants in this transitive action.

Stability Theory

Definitions of Stability

In geometric invariant theory, consider a reductive algebraic group G acting on a projective variety X over an algebraically closed field, equipped with a linearized ample line bundle L. A point x \in X is semistable if there exists some positive integer m > 0 and a G-invariant section s \in H^0(X, L^{\otimes m})^G such that s(x) \neq 0. Equivalently, in the affine cone over X, the orbit closure \overline{G \cdot \hat{x}} (where \hat{x} is a lift of x) does not contain the origin. The set of semistable points, denoted X^{ss}(L), forms an open subset of X. A semistable point x \in X^{ss}(L) is stable if its stabilizer G_x is finite (i.e., \dim G_x = 0) and its orbit G \cdot x is closed in X^{ss}(L). The set of stable points, denoted X^s(L), is an open subset of X^{ss}(L). A semistable point x \in X^{ss}(L) is polystable if its orbit G \cdot x is closed in X^{ss}(L). Thus, every stable point is polystable, but polystable points may have positive-dimensional reductive stabilizers. The polystable locus X^{ps}(L) is the set of fixed points of the GIT quotient map on X^{ss}(L). A semistable point is properly semistable if it is not stable; these points have orbits that are not closed, but their closures contain polystable points.

Hilbert-Mumford Numerical Criterion

The Hilbert-Mumford numerical criterion provides a concrete method to determine the stability of points in the context of a reductive group action on a projective variety equipped with a linearized ample line bundle. Central to this criterion are one-parameter subgroups of the group G, which are algebraic homomorphisms \lambda: \mathbb{C}^* \to G. These subgroups serve to probe the behavior of orbits under the group action by examining limits of the form \lim_{t \to 0} \lambda(t) \cdot x for a point x in the variety, thereby revealing instability through "escape to infinity" in the projective setting. For a point x \in X (where X is the projective variety) and a one-parameter subgroup \lambda, the numerical function \mu(x, \lambda) is defined using a lift \tilde{x} of x to the total space of the line bundle L. Specifically, the action of \lambda decomposes the fiber L_x into weight spaces with integer weights determined by the cocharacter associated to \lambda, and \mu(x, \lambda) is the minimum of these weights (up to normalization by the degree of the linearization). This function quantifies the "slope" of the orbit under \lambda, with negative values indicating that the limit as t \to 0 does not exist or lies outside the affine cone, signaling instability.^[2] The criterion states that a point x is semistable if and only if \mu(x, \lambda) \geq 0 for every one-parameter subgroup \lambda of G. Equivalently, the closure of the orbit \overline{G \cdot x} intersects the zero section of the line bundle if and only if there exists some \lambda with \mu(x, \lambda) < 0. For proper (or strict) stability, the condition strengthens to \mu(x, \lambda) > 0 for all nontrivial one-parameter subgroups \lambda, ensuring that the limit \lim_{t \to 0} \lambda(t) \cdot x does not exist for any such \lambda, in addition to the orbit having finite stabilizer. This numerical test reduces the global stability question to checking weights against a lattice of cocharacters, making it computationally tractable.^[17]^[2] A classic example illustrates the criterion in the action of G = \mathrm{SL}(2, \mathbb{C}) on the projective space \mathbb{P}^n parameterizing binary forms of degree n, via the representation on \mathrm{Sym}^n(\mathbb{C}^2)^*. Consider a one-parameter subgroup \lambda_r(t) = \begin{pmatrix} t^r & 0 \\ 0 & t^{-r} \end{pmatrix} for r > 0. For a form f = \sum_{k=0}^n a_k x^{n-k} y^k, the action yields \lambda_r(t) \cdot f = \sum_{k=0}^n a_k t^{r(n - 2k)} x^{n-k} y^k, so the weights are r(n - 2k) for terms with a_k \neq 0. Thus, \mu(f, \lambda_r) = r \cdot \min \{ n - 2k \mid a_k \neq 0 \}. The form f is semistable if and only if this minimum is at least 0 for all r > 0, i.e., no term with k > n/2, meaning no root at [0:1] with multiplicity exceeding n/2. For instance, when n=4, monomials like x^4 (k=0, weight 4r >0) or x^2 y^2 (k=2, weight 0) are semistable, while y^4 (k=4, weight -4r <0) is unstable.^[2]^[18]

Construction of GIT Quotients

Proj Construction and Invariant Rings

In geometric invariant theory, the Proj construction yields the algebraic quotient of a projective variety under a linearized group action by taking the projective spectrum of the associated graded invariant ring. Consider a projective variety X over an algebraically closed field k, equipped with an action of a reductive algebraic group G and an ample line bundle L on X with a G-linearization \theta: G \times L \to L. The invariant ring is the graded k-algebra

R = \bigoplus_{n \geq 0} H^0(X, L^{\otimes n})^G,

comprising all G-invariant global sections of the tensor powers of L.^[19] Hilbert's finiteness theorem asserts that R is finitely generated as a k-algebra when G is reductive, a result that extends classical invariant theory to this setting via properties of the Reynolds operator and Nagata's criterion for linear reductivity.^[19] This finite generation ensures that \operatorname{Proj}(R) is a well-defined projective scheme over k, serving as the GIT quotient X //_\theta G. The construction parameterizes the semistable locus X^{\theta\text{-}ss} \subseteq X, where points are those admitting a non-zero invariant section in some power of L.^[19]^[2] The canonical morphism \phi: X^{\theta\text{-}ss} \to \operatorname{Proj}(R) projects semistable points to their orbit classes, with \phi(x) = \phi(y) if and only if the closures of the G-orbits through x and y intersect in X^{\theta\text{-}ss}. This map satisfies the universal property for categorical quotients: any G-invariant morphism f: X^{\theta\text{-}ss} \to Z to a projective scheme Z factors uniquely as f = \psi \circ \phi for some morphism \psi: \operatorname{Proj}(R) \to Z.^[19]^[2] As a categorical quotient, \phi is G-invariant, separates closed orbits in X^{\theta\text{-}ss}, and identifies points precisely when their orbit closures meet, yielding fibers that are the G-orbits themselves over a dense open subset corresponding to stable points. This structure embeds the orbit space of stable points as an open subscheme of \operatorname{Proj}(R), providing a projective compactification.^[19]^[2]

Properties and Geometric Interpretation

In geometric invariant theory (GIT), the quotient map \pi: X^{\mathrm{ss}} \to Y from the semistable locus X^{\mathrm{ss}} to the GIT quotient Y = \mathrm{Proj}(R(X,L)^G) satisfies the properties of a good quotient for a reductive group action. Specifically, \pi is a projective morphism that is G-invariant, meaning it is constant on G-orbits and maps G-invariant open subsets of X^{\mathrm{ss}} to open subsets of Y; moreover, the fibers of \pi consist of unions of G-orbits, with each fiber containing a unique closed orbit.^[1]^[2] On the stable locus X^{\mathrm{s}} \subset X^{\mathrm{ss}}, the restriction \pi|_{X^{\mathrm{s}}}: X^{\mathrm{s}} \to Y^{\mathrm{s}} (where Y^{\mathrm{s}} is the image) forms a geometric quotient. This means it retains the good quotient properties while additionally separating distinct closed orbits—points in distinct closed orbits map to distinct points in Y^{\mathrm{s}}—and is an affine morphism over its image, providing a faithful geometric realization of the orbit space for stable points.^[1]^[2] However, the presence of semistable points that are not stable means that the quotient map \pi is only a categorical quotient globally, not geometric, as multiple closed orbits may be identified if their closures intersect at properly semistable points. For instance, consider the action of G = \mathbb{C}^\times on the affine space X = \mathbb{C}^2 by t \cdot (x_1, x_2) = (t^{-1} x_1, t^{-1} x_2); the invariant ring is \mathbb{C}, yielding a quotient that is a single point, with all points semistable and none stable, so all orbits are identified in the quotient.^[2]^[16] A key theorem ensures the geometric robustness of GIT quotients in the projective setting: if X is a projective variety over an algebraically closed field and L is a G-linearized ample line bundle, then the GIT quotient Y is itself a projective variety. This projectivity follows from the Proj construction applied to the graded ring of invariants, guaranteeing that Y embeds as a closed subscheme in a projective space.^[1]^[2]

Advanced Topics and Applications

Relation to Other Quotient Constructions

Symplectic reduction offers a parallel quotient construction to GIT in the setting of Hamiltonian actions of compact Lie groups on symplectic manifolds. For a compact group K acting Hamiltonially on a symplectic manifold (M, \omega) with moment map \mu: M \to \mathfrak{k}^*, the symplectic quotient at level \xi \in \mathfrak{k}^* is defined as \mu^{-1}(\xi)/K, provided the level set is smooth and the action is free there. This yields a reduced symplectic manifold, capturing the invariants of the action in a geometric manner. Kirwan's convexity theorem asserts that, for torus actions on compact symplectic manifolds, the image \mu(M) is a convex polytope, mirroring the numerical stability conditions in GIT where weights lead to non-negative Hilbert-Mumford criteria \mu \geq 0. The Kempf-Ness theorem bridges GIT and symplectic reduction by relating algebraic stability to symplectic geometry. For a complex reductive group G = K^\mathbb{C} acting linearly on a projective variety X \subset \mathbb{P}(V) with a G-linearized ample line bundle, an orbit G \cdot x is GIT-polystable if and only if it intersects the zero level of the moment map \mu: X \to \mathfrak{k}^* associated to a compatible Kähler metric on X, and the unique point in the intersection is fixed by the K-stabilizer. This equivalence shows that GIT semistable points correspond to those with bounded moment map norm \|\mu(x)\|, providing a geometric interpretation of instability via the failure to minimize this norm. Consequently, for projective representations, the GIT quotient X^{ss} // G is homeomorphic to the symplectic quotient \mu^{-1}(0)/K over the smooth locus. Topological quotients, as explored in Borel's foundational work on linear algebraic group actions, consider the orbit space X/G equipped with the quotient topology for an action on a topological space X. For affine varieties under reductive group actions, this coincides with the categorical quotient \operatorname{Spec}(k[X]^G), inheriting an algebraic structure as an affine variety. However, for projective varieties, the topological quotient lacks a natural algebraic or projective structure, as invariant rings may not be finitely generated, necessitating GIT's stability conditions to produce a projective moduli space.

Moduli Spaces in Algebraic Geometry

One of the primary applications of geometric invariant theory (GIT) lies in the construction of compact moduli spaces for algebraic objects, where the quotient construction provides a projective variety parametrizing isomorphism classes of stable objects. In particular, GIT addresses the challenge of compactifying coarse moduli spaces by incorporating appropriate stability conditions that allow for limits of degenerating families. This approach ensures the resulting space is proper and algebraic, facilitating the study of geometric properties and deformation theory.^[20] A seminal example is the Deligne-Mumford compactification of the moduli space of genus g curves with n marked points, denoted \overline{\mathcal{M}}_{g,n}. Using GIT, Deligne and Mumford constructed this space as the quotient of the Hilbert scheme of stable pointed curves by the action of the projective linear group, where stability requires that the curve has only nodal singularities and that the marked points impose independent conditions on the canonical embedding for sufficiently high multiples. This compactification includes stable curves, which are nodal curves with finite automorphism groups, providing a projective variety that extends the open moduli space \mathcal{M}_{g,n} of smooth curves. The construction relies on Hilbert stability for the pluri-canonical linearization, ensuring the quotient is a coarse moduli space with good geometric interpretation.^[20]^[21] An illustrative concrete case is the GIT quotient for the moduli space of binary quartics under the action of \mathrm{SL}(2,\mathbb{C}). The space of binary quartics, which are homogeneous polynomials of degree 4 in two variables, is \mathbb{P}^4, and the quotient by the natural action yields \mathbb{P}^1 as the moduli space, parametrizing isomorphism classes of such forms up to scalar. This quotient demonstrates how GIT resolves the semi-stable locus to produce a simple projective curve, with unstable points corresponding to forms with excessive roots, and it serves as a model for understanding j-invariants in elliptic curve moduli.^[22] GIT also constructs moduli spaces of semistable vector bundles on a smooth projective curve C of genus g \geq 2. By embedding the moduli problem into the Hilbert scheme of points or using the determinant line bundle construction, one forms a projective scheme on which \mathrm{PGL}(r) acts, where r is the rank; the GIT quotient of the semistable locus then parametrizes S-equivalence classes of semistable bundles of fixed rank and degree coprime to the rank. This yields a projective variety whose points correspond to polystable bundles, providing a compactification that includes direct sums of stable bundles in the boundary. The stability condition ensures boundedness, and the construction via the Hilbert-Mumford criterion identifies semistable bundles as those without destabilizing subbundles.^[23] However, the choice of linearization in GIT quotients introduces limitations, as different ample linearizations can alter the semistable locus, leading to wall-crossing phenomena where the moduli space jumps across walls in the parameter space of linearizations. For instance, in the construction of \overline{\mathcal{M}}_{g,n}, varying the multiple of the pluri-canonical bundle changes stability conditions, resulting in birational modifications like flips or blow-ups that refine the compactification. These wall-crossings allow for a family of related moduli spaces, each suitable for different geometric questions, but require careful analysis to relate them via VGIT (variation of GIT) maps.^[12]

References

[1]
Geometric Invariant Theory - SpringerLink
Free delivery 14-day returnsOct 29, 2012 · “Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is ...
[2]
[PDF] introduction to geometric invariant theory - Yale Math
It is, basically, a consequence of the following theorem. Theorem 3.2 (Hilbert-Mumford Criterion). Let G be a reductive algebraic group acting rationally on a.
[3]
geometric invariant theory in nLab
Jul 19, 2021 · Geometric invariant theory studies the construction of moduli spaces / moduli stacks in terms of quotients / action groupoids.
[4]
[PDF] LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
The chapter concludes with a brief discussion of the algebraic relationships, known as "syzygies", that exist among the fundamental invariants and covariants.
[5]
[PDF] Sturmfels, Algorithms in invariant theory
Jan 4, 2013 · The aim of this monograph is to provide an introduction to some fundamental problems, results and algorithms of invariant theory. The focus will ...
[6]
[PDF] An Introduction to Invariants and Moduli (Cambridge Studies in ...
(3) I have included the Cayley-Sylvester formula in order to compute explic- itly the Hilbert series of the classical binary invariant ring since I believe.
[7]
[PDF] The Fundamental Theorem of Invariant Theory
Nov 15, 2006 · 1890: Hilbert's finiteness theorem: C[V]G is finitely generated for a wide class of groups (linearly reductive groups). Gordan: “Das ist ...
[8]
[PDF] Invariants: Computation and Applications - arXiv
Dec 17, 2024 · Hilbert's work ended the classical period of invariant theory and stimulated the rapid development of new fields of modern algebra, ...
[9]
On the 14-th Problem of Hilbert - jstor
writer noticed that the first example is also a counter-example to the original. 14-th problem (and the example will be stated in the present paper). By ...
[10]
[PDF] arXiv:2302.14499v1 [math.AG] 28 Feb 2023
Feb 28, 2023 · Mumford's geometric invariant theory (GIT) [74] for reductive groups provides a method for constructing quotients of reductive group actions in ...
[11]
[PDF] geometric invariant theory and symplectic quotients
We briefly summarise the main results in affine and projective GIT below. If X ⊂ An is an affine variety over an algebraically closed field k which is cut out ...Missing: issues | Show results with:issues
[12]
[PDF] MODULI SPACES AND GEOMETRIC INVARIANT THEORY
More generally, given a scheme X with a G-linearisation L,. Mumford defines a GIT quotient using invariant sections of positive powers of L whose non- vanishing ...<|control11|><|separator|>
[13]
[PDF] An Introduction to Invariants and Moduli (Cambridge Studies in ...
Theorem 5.3 (Nagata, Mumford). Suppose that a linearly reductive group G ... By the Hilbert-Mumford Criterion (7.3), instability means that the.
[14]
[PDF] Reductive Groups
Abstract. These notes are a guide to algebraic groups, especially reductive groups, over a field. Proofs are usually omitted or only sketched.
[15]
[PDF] Introduction to actions of algebraic groups
They present some fundamental results and examples in the theory of algebraic group actions, with special attention to the topics of geometric invariant theory ...
[16]
[PDF] Course Notes for Math 780-1 (Geometric Invariant Theory)
Definition: If G acts on a quasiprojective variety X, then a categorical quotient of X by the action of G is a pair consisting of a quasiprojective Y and a ...
[17]
[PDF] An elementary proof of the Hilbert-Mumford criterion - ISI Bangalore
Dec 29, 1999 · A classical result of geometric invariant theory is the Hilbert-Mumford semistability criterion. In one form, it deals with a linear action.Missing: original | Show results with:original<|control11|><|separator|>
[18]
[PDF] Geometric Invariant Theory Math 533 - Spring 2011 - Pooya Ronagh
Definition 3. A linear algebraic group is an algebraic group G that is an affine variety. Right away this tells you something about the coordinate ring of it!
[19]
[PDF] Moduli Problems and Geometric Invariant Theory.
Abstract. In this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.
[20]
[PDF] The irreducibility of the space of curves of given genus
We will study in this section three aspects of the theory of stable curves: their pluri-canonical linear systems, their deformations, and their automorphisms.
[21]
[PDF] GIT Constructions of Moduli Spaces of Stable Curves and Maps - arXiv
Abstract. Gieseker's plan for using GIT to construct the moduli spaces of stable curves, now over 30 years old, has recently been extended to moduli.
[22]
[PDF] Stack structures on GIT quotients parametrizing hypersurfaces
Inside the moduli space of binary sextics. 8. CUBIC CURVES AND SURFACES. In this section we describe the relation between the GIT quotient scheme and the stacky.
[23]
[PDF] 4. Vector Bundles on a Smooth Curve. - University of Utah Math Dept.
In this section, we will construct projective moduli spaces for semistable vector bundles on a smooth projective curve C using GIT. The construction we present ...