Projective K3 Surfaces
If L is a line bundle on a K3 surfaces, then the curves in the linear system have genus g where c12(L) =2g-2. A K3 surface with a line bundle L like this is called a K3 surface of genus g. A K3 surface may have many different line bundles making it into a K3 surface of genus g for many different values of g. The space of sections of the line bundle has dimension g+1, so there is a morphism of the K3 surface to projective space of dimension g. There is a moduli space Fg of K3 surfaces with a primitive ample line bundle L with c12(L) =2g-2, which is nonempty of dimension 19 for g≥ 2. Mukai (2006) showed that this moduli space Fg is unirational if g≤13, and V. A. Gritsenko, Klaus Hulek, and G. K. Sankaran (2007) showed that it is of general type if g≥63. Voisin (2008) gave a survey of this area.
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