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A smooth complete algebraic surface S is of type K3 if S is regular and the canonical class KS is trivial. A primitively polarized K3 surface is a pair (S, h) of a K3 surface S and a primitive ample divisor class h ∈ PicS. The integer g := 12 (h) + 1 ≥ 2 is called the genus of (S, h). The moduli space of primitively polarized K3 surfaces of genus g exists as a quasi-projective variety, which we denote by Fg. As is well known a general polarized K3 surface of genus 2 ≤ g ≤ 5 is a complete intersection of hypersurfaces in a weighted projective space: (6) ⊂ P(1112), (4) ⊂ P, (2) ∩ (3) ⊂ P and (2) ∩ (2) ∩ (2) ⊂ P. In connection with the classification of Fano threefolds, we have studied the system of defining equations of the projective model S2g−2 ⊂ P and shown that a general polarized K3 surface of genus g is a complete intersection with respect to a homogeneous vector bundle Vg−2 in a g-dimensional Grassmannian G(n, r), g = r(n − r), in a unique way for the following six values of g:

Polarized K3 surfaces of genus thirteen