Fix a smooth projective algebraic curve (over ) of genus . Unlike the case of line bundles, it has been observed that in general all vector bundles are not classifiable. For example, one can construct a family of vector bundles of rank parametrized by the disk where all the fibers apart from the origin are mutually isomorphic, but not isomorphic to the fiber at the origin ([1, 11.32]). This "jump phenomenon" illustrates that the set of all vector bundles on curves is not even separated. In other words, even the coarse moduli space does not exist. To construct well-behaved moduli spaces of vector bundles, Mumford's geometric invariant theory hints at finding stable conditions on vector bundles and restricting our attention to stable vector bundles.
By definition, every line bundle is stable. The following implications partially explain that why stable bundles are "nice".
The moduli space of stable vector bundles over of rank and degree was first given by Mumford  and Seshadri . Later, Gieseker gave a different construction which generalized to higher dimensions. Simpson invented a more natural and general method using Grothendieck's Quot scheme which also extends to singular curves and higher dimensions (see ).
is simply the Picard variety we have constructed. We have the natural map sending to its determinant bundle . We fix a line bundle and study the fiber of this map. 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. 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 . Denote to be set of matrices having rank over . is a subvariety of and the image of the above map lies in it.
To apply GIT to construct , we need to study the stability of under the action of . This consists of two steps: for ,
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.
So we have shown:
In fact, the converse is also true:
We omit the proof of this fact since it is a bit long (see [1, 10.70]). 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.
Now let us come to the second step.
Let us first show . Since is semistable, we know that by the last lemma. Suppose , then by Serre duality, we have a nonzero morphism , which gives a map . Let , then as . So is unstable, a contradiction.
Next we need to show that . The map is skew-symmetric and vanishes on , hence induces a sheaf morphism . From we know that . So .
Finally we conclude is semistable as is the Gieseker point of generated by global sections and . ¡õ
is smooth and if is nonempty (when , or , coprime, or , ), then . Moreover, is a fine moduli space if and only if are coprime. The dimension of is as expected. In particular, if it is nonempty.
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