# Geometric invariant theory and flips

@article{Thaddeus1994GeometricIT, title={Geometric invariant theory and flips}, author={Michael Thaddeus}, journal={Journal of the American Mathematical Society}, year={1994}, volume={9}, pages={691-723} }

We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and explain the relationship with the minimal model programme. Moreover, we express the flip as the blow-up and blow-down of specific ideal sheaves, leading, under certain hypotheses, to a quite explicit description of the flip. We apply these ideas to various familiar moduli problems, recovering results of… Expand

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#### References

SHOWING 1-10 OF 44 REFERENCES

Stable pairs, linear systems and the Verlinde formula

- Mathematics
- 1992

We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the… Expand

Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31

- Mathematics
- 1984

These notes describe a general procedure for calculating the Betti numbers of the projective quotient varieties that geometric invariant theory associates to reductive group actions on nonsingular… Expand

Variation of geometric invariant theory quotients

- Mathematics
- 1994

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of… Expand

Moduli of vector bundles on curves with parabolic structures

- Mathematics
- 1980

Let H be the upper half plane and T a discrete subgroup of AutH. Suppose that H mod Y is of finite measure. This work stems from the question whether there is an algebraic interpretation for the… Expand

On the variation in the cohomology of the symplectic form of the reduced phase space

- Mathematics
- 1982

is called the momentum mapping of the Hamiltonian T-action. Given (1.1), the condition (1.2) just means that T acts along the fibers of J. For the basic definitions and properties of non-commutative… Expand

Parabolic vector bundles on curves

- Mathematics
- 1989

Seshadri introduced the notion of parabolic structures on vector bundles [4] and later constructed a moduli space for semistable parabolic vector bundles on curves [2]. In this small note we describe… Expand

Quotient Spaces Modulo Reductive Algebraic Groups

- Mathematics
- 1972

This conjecture is known to be true when the base field is of characteristic zero. In fact we can then find a linear F and this is an immediate consequence of the complete reducibility of finite… Expand

Symmetric products of an algebraic curve

- Mathematics
- 1962

LET X be a nonsingular irreducible complete algebraic curve over the field C of complex numbers, and let X(n) denote the nth symmetric product of X. The first part of this paper is devoted to an… Expand

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

- Mathematics
- 1993

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure… Expand

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES II

- Mathematics
- 1991

In this paper we continue our investigation of the moduli space of stable pairs introduced in Part I. We obtain certain topological information, and we give a proof that this moduli space admits the… Expand