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## Geometric Invariant Theory for Polarized Curves

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Marius Ghergu. Linear Second Order Elliptic Operators. Progress in Partial Differential Equations. Michael Reissig. Invariants for homology 3-spheres Computational invariant theory Reflection groups and invariant theory Lecture notes on Chern-Simons-Witten theory Invariants for pattern recognition and classification Classical invariant theory Classical and involutive invariants of Krull domains Selected papers on harmonic analysis, groups, and invariants Representations and invariants of the classical groups Quantum groups and knot invariants Algebraic homogeneous spaces and invariant theory Equivalence, invariants, and symmetry Invariance theory, the heat equation, and the Atiyah-Singer index theorem Invariant potential theory in the unit ball of Cn Temperley-Lieb recoupling theory and invariants of 3-manifolds Normally hyperbolic invariant manifolds in dynamical systems Algebraic geometry IV Geometric invariant theory Applications of Invariance in Computer Vision Quantum invariants of knots and 3-manifolds Invariant function spaces on homogeneous manifolds of Lie groups and applications Invariant distances and metrics in complex analysis Polynomial invariant of finite groups Theory of algebraic invariants Groups, generators, syzygies and orbits in invariant theory Lie groups, their discrete subgroups, and invariant theory Geometric invariance in computer vision The differential invariants of generalized spaces Topics in invariant theory Topics in Invariant Theory Invariant measures on groups and their use in statistics Operator algebras, unitary representations, enveloping algebras, and invariant theory Invariant theory and tableaux Algebraische Transformationsgruppen und Invariantentheorie Topological methods in Galois representation theory Group invariance applications in statistics Group actions and invariant theory Decomposition and invariance of measures, and statistical transformation models The method of equivalence and applications Invariant theory and superalgebras Relative invariants of sheaves Geometrische Methoden in der Invariantentheorie Relative invariants of rings Lectures on moduli of curves Young tableaux in combinatorics, invariant theory, and algebra Invariance and system theory Strong approximations in probability and statistics Algebraic structures of knot modules Curvature and characteristic classes Meromorphe Differentialgleichungen Invariant manifolds Invariant variational principles Invariants for real-generated uniform topological and algebraic categories Homotopy invariant algebraic structures on topological space Broken scale invariance and the light cone The theory of determinants, matrices, and invariants Eddy Campbell.

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The simple-minded idea of an orbit space. There is in fact no general reason why equivalence relations should interact well with the rather rigid regular functions polynomial functions , which are at the heart of algebraic geometry.

The direct approach can be made, by means of the function field of a variety i. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put it in the Preface to the book:. In Chapter 5 he isolates further the specific technical problem addressed, in a moduli problem of quite classical type — classify the big 'set' of all algebraic varieties subject only to being non-singular and a requisite condition on polarization.

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## Geometric Invariant Theory for Polarized Curves | Gilberto Bini | Springer

The moduli are supposed to describe the parameter space. For example, for algebraic curves it has been known from the time of Riemann that there should be connected components of dimensions. In the coarse moduli problem Mumford considers the obstructions to be:. It is the third point that motivated the whole theory.

As Mumford puts it, if the first two difficulties are resolved. To deal with this he introduced a notion in fact three of stability. This enabled him to open up the previously treacherous area — much had been written, in particular by Francesco Severi , but the methods of the literature had limitations. The birational point of view can afford to be careless about subsets of codimension 1.

To have a moduli space as a scheme is on one side a question about characterising schemes as representable functors as the Grothendieck school would see it ; but geometrically it is more like a compactification question, as the stability criteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as moduli space: varieties can degenerate to having singularities. On the other hand, the points that would correspond to highly singular varieties are definitely too 'bad' to include in the answer.

The correct middle ground, of points stable enough to be admitted, was isolated by Mumford's work.

The concept was not entirely new, since certain aspects of it were to be found in David Hilbert 's final ideas on invariant theory, before he moved on to other fields. The book's Preface also enunciated the Mumford conjecture , later proved by William Haboush.