MSC:

14D20 Algebraic moduli problems, moduli of vector bundles, {For analytic moduli problems, See 32G13}

14D22 Fine and coarse moduli spaces

14D25 Geometric invariants, See also {14L30}

In \S 1 of Faltings' article \cite{Fal} a ``GIT-free'' construction is given for the moduli spaces of vector bundles on curves using generalized theta functions. Incidentally, this construction is implicitly described in Le Potier's article \cite{LP2}. The aim of this paper is to generalize the {\em duality construction} to projective surfaces.

For a rank two vector bundle $E$ on the projective plane $\pdop^2$, the divisor $D_E$ of its jumping lines is a certain generalization of the Chow divisor of a projective scheme. We give a generalization of this divisor for coherent sheaves on surfaces. Using this duality we construct the moduli space of coherent sheaves on a surface that does not use Mumford's geometric invariant theory. Furthermore, we obtain a finite morphism from this moduli space to a linear system, which generalizes the divisors of jumping lines. Applying this construction to curves, we get exactly Faltings' construction. The moduli space we construct here can also be obtained by using GIT. This construction is carried out in \S8.2 of the book \cite{HL} of Huybrechts and Lehn. Le Potier obtained this moduli space in \cite{LP1} for surfaces with ``many lines'' (see \S \ref{SBAR} for an exact definition). However, it is the modest hope of the author that the construction presented here provides new insight into the geometry of moduli spaces.

First we outline this concept, which generalizes the famous {\em strange duality} to moduli of coherent sheaves on surfaces. To do so we define duality between schemes in part \ref{SDUAL}, giving three examples of ``natural dualities''. In section \ref{SDCON} the duality construction is given. In order to avoid a too technic presentation of the construction itself, we defer the proofs to the following section. The last section is dedicated to the Barth morphism. In order to simplify the discussion we restrict ourselves to moduli spaces of sheaves of rank two with trivial determinant. The interested reader will be able to extend this to arbitrary rank and determinant.

The author is thankful to his thesis advisor, H.~Kurke, for many fruitful discussions.
Keywords: duality construction, generalized theta functions, moduli of sheaves, strange duality