# An Introduction to Invariants and Moduli Garcia bath. Algebraic Geom. MR  Igor V. Dolgachev and Yi Hu , Variation of geometric invariant theory quotients , Inst. With an appendix by Nicolas Ressayre. Translated from the and Japanese editions by W.

## Geometric invariant theory and applications to moduli problems

MR  D. Mumford , J. Fogarty , and F. Kirwan , Geometric invariant theory , 3rd ed. Differential Geom. Automorphisms of forms , Math. In other words, we are going to construct the space of stable vector bundles using GIT. Similarly to the case of Picard varieties, we will assume so that Riemann-Roch brings us some convenience. Now let us concentrate on the case.

Similarly to the case of Picard varieties, we will associate to each isomorphism class of vector bundles of rank 2 a -orbit of a matrix and study its stability. Let be a vector bundle of rank 2 with and generated by global sections.

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Let and be a basis. Since is generated by global sections, we have a surjection The pairing induces a map which is injective since is generated by global sections. The composition map is given by the matrix Let be the set of skew-symmetric matrices with entries in a vector space. Then and we call it the Gieseker point of. The different choices of the marking correspond to the -orbit of under the action Moreover, is isomorphic to the image of , hence one can recover from its Gieseker points.

So we have proved:. Since is a vector bundle of rank 2, the matrix have rank 2 over the function field.

## An Introduction to Invariants and Moduli by Shigeru Mukai, W. M. Oxbury - teweepreli.ga

Denote to be set of matrices having rank over. To apply GIT to construct , we need to study the stability of under the action of. Step 1 is semistable stable if and only if is semistable stable. Step 2 Every semistable is the Gieseker point for some. Assuming these two steps, we can construct as a GIT quotient immediately. For the first step, we need the following observation. This phenomenon did not appear in the case of line bundles.

We omit the proof of this fact since it is a bit long see [1, The key idea is to construct -semi-invariant polynomials using the Pfaffians or the radical vectors of skew symmetric matrices depending on whether is even or odd. The condition for -semistability is already quite similar to semistability.

One can use Riemann-Roch to prove the following, which finishes the proof of the first step. Let us first show. Since is semistable, we know that by the last lemma.

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Suppose , then by Serre duality, we have a nonzero morphism , which gives a map. So is unstable, a contradiction.

Next we need to show that. The map is skew-symmetric and vanishes on , hence induces a sheaf morphism.

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From we know that. Finally we conclude is semistable as is the Gieseker point of generated by global sections and. Moreover, is a fine moduli space if and only if are coprime. The dimension of is as expected. In particular, if it is nonempty. Mumford, Projective invariants of projective structures and applications, Proc.

## Geometric Invariant Theory

Definition 1 The slope of a vector bundle is the ratio. Equivalently, is stable resp. Proposition 1 Stable simple indecomposable. Proof The definition of stability implies that every nonzero endomorphism is an isomorphism, hence is a scalar by looking at one fiber. So there is no nonzero morphism as. Hence by Serre duality. By the exact sequence We know that is surjective. By Nakayama's lemma we know that is surjective. Proposition 4 Sending to the -orbit of its Gieseker points gives an injection. Proof The only thing to check is that the orbits in are free. We then have the following commutative diagram This gives an endomorphism of. Since is stable, we know that is simple. Hence with. Namely , which acts trivially on.

Lemma 1 If is semistable stable , then for any sub-line bundle , we have. An Introduction to Invariants and Moduli An Introduction to Invariants and Moduli An Introduction to Invariants and Moduli An Introduction to Invariants and Moduli An Introduction to Invariants and Moduli

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