A geometric invariant theory construction of moduli spaces of stable maps

We construct the moduli spaces of stable maps, Graphic, via geometric invariant theory (GIT). This construction is only valid over Graphic, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, Graphic; this is valid over Graphic. In another paper by the first author, a small part of the argument is replaced, making the result valid in far greater generality. Our method follows the one used in the case n = 0 by Gieseker in [9], 1982, Lectures on Moduli of Curves to construct Graphic, though our proof that the semistable set is nonempty is entirely different.