that every unirational K3 surface over an algebraically closed field must have Picard number 22. (In particular, a unirational K3 surface must be supersingular.) Conversely, Artin conjectured that every K3 surface with Picard number 22 must be unirational. Artin's conjecture was proved in characteristic 2 by . Proofs in every characteristic p at least 5 were claimed by and , but later refuted by . History The first example of a K3 surface with Picard number 22 was given by , who observed that the Fermat quartic w4 + x4 + y4 + z4 = 0 has Picard number 22 over algebraically closed fields of characteristic 3 mod 4. Then Shioda showed that the elliptic modular surface of level 4 (the universal generalized elliptic curve E(4) → X(4)) in characteristic 3 mod 4 is a K3 surface with Picard number 22, as is the Kummer surface of the product of two supersingular elliptic curves in odd characteristic. showed that all K3 surfaces with Picard number 22 are double covers of the projective plane. In the case of characteristic 2 the double cover may need to be an inseparable covering. The discriminant of the intersection form on the Picard group of a K3 surface with Picard number 22 is an even power p2e of the characteristic p, as was shown by Artin and Milne. Here e is called the Artin invariant of the K3 surface. Artin showed that 1 ≤ e ≤ 10. There is a corresponding Artin stratification of the moduli spaces of supersingular K3 surfaces, which have dimension 9. The subspace of supersingular K3 surfaces with Artin invariant e has dimension e − 1. Examples In characteristic 2, z2 = f(x, y) , for a sufficiently general polynomial f(x, y) of degree 6, defines a surface with 21 isolated singularities. The smooth projective minimal model of such a surface is a unirational K3 surface, and hence a K3 surface with Picard number 22. The largest Artin invariant here is 10. Similarly, in characteristic 3, z3 = g(x, y) , for a sufficiently general polynomial g(x, y) of degree 4, defines a surface with 9 isolated singularities. The smooth projective minimal model of such a surface is again a unirational K3 surface, and hence a K3 surface with Picard number 22. The highest Artin invariant in this family is 6. described the supersingular K3 surface in characteristic 2 with Artin