Georg Hein: Faltings' Construction of the Moduli Space of Vector Bundles

We will give Faltings' construction of the moduli space of semistable
vector bundles on a smooth projective curve. A central idea is to use the generalized Theta line bundle which turns out to be an ample line bundle on the moduli space.
Furthermore, we show why we have to add S-equivalence classes to obtain a projective moduli space. We will see that semistability of a sheaf E  on a curve X,  is equivalent to the existence of a nontrivial sheaf F,  such that the tensor product E \otimes F  has no cohomology. Once we have this, the construction is short and elegant. At the end we show how this construction can be used to obtain the moduli space of Mumford-Takemoto semistable sheaves on algebraic surfaces.

Literature:

• G. Faltings, Stable G-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507-68.

• G. Hein, Duality construction of moduli spaces, Geom. Dedicata 75 (1999),101--113.

• C.S. Seshadri, Vector bundles on curves in Linear algebraic groups and their representations (Los Angeles, CA, 1992), 163--200,  Contemp. Math. 153, Amer. Math. Soc., Providence, RI, 1993.