Supersingular K3 surface

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In algebraic geometry, a supersingular K3 surface is a particular type of K3 surface. Such an algebraic surface has its cohomology generated by algebraic cycles; in other words, since the second Betti number^[1] of a K3 surface is always 22, such a surface must possess 22 independent elements in its Picard group (ρ = 22).

Such surfaces can exist only in positive characteristic, since in characteristic zero Hodge theory gives an upper bound of 20 independent elements in the Picard group. In fact the Hodge diamond for any complex K3 surface is the same (see classification) and the middle row reads 1, 20, 1. In other words h^2,0 and h^0,2 both take the value 1, with h^1,1 = 20. Therefore the dimension of the space spanned by the algebraic cycles is at most 20 (in characteristic zero).

1 History
2 Examples
3 Kummer surfaces
4 Notes
5 External link

[edit] History

These surfaces were first dicovered by André Weil and John Tate and then more fully developed by Michael Artin and Tetsuji Shioda. It has been conjectured, by Artin, that every supersingular K3 surface is unirational; this conjecture remains open as of 2007. Shioda has shown that supersingular K3 surfaces are double covers of the projective plane.^[2] In the case of characteristic 2 the double cover may need to be an inseparable covering.

Artin's conjecture has been shown to be true in characteristic two by Shafarevich and Rudakov^[3] and more recently by Shimada. The conjecture remains open in characteristic three; several families of examples have been constructed, showing that it is at least plausible.

A supersingular K3 surface is also a Calabi-Yau manifold, in positive characteristic, and is perhaps of some interest to physicists as well as algebraic geometers.

The discriminant of the intersection form on the Picard group of a supersingular K3 surface is an even power

p^2e

of the characteristic p, as was shown by Michael Artin and James S. Milne. Here e is defined to be the Artin invariant. We have

1 ≤ e ≤ 10

as was also shown by Michael Artin. There is a corresponding Artin stratification.

[edit] Examples

In characteristic two,

z² = f(x, y) ,

for a sufficiently general polynomial f(x, y) of degree six, defines a surface with twenty-one isolated singularities. It can be shown that the smooth projective minimal model of the function field of such a surface is a supersingular K3 surface. The largest Artin invariant here is ten.

Similarly, in characteristic three,

z³ = g(x, y) ,

for a sufficiently general polynomial g(x, y) of degree four, defines a surface with nine isolated singularities.It can be shown that the smooth projective minimal model of the function field of such a surface is again a supersingular K3 surface. The highest Artin invariant in this family is six.

In characteristic five, Ichiro Shimada and Duc Tai Pho have recently demonstrated^[4] that every supersingular K3 surface with Artin invariant less than four is birationally equivalent to a surface with equation

z⁵ = h(x, y) ,

and thus every such surface is unirational.

[edit] Kummer surfaces

If the characteristic p is greater than 2, all supersingular K3 surfaces S with Artin invariant 0, 1 and 2 are birationally Kummer surfaces, in other words quotients of an abelian surface A by the mapping x → − x. More precisely, A should be a supersingular abelian surface, isogenous to a product of two supersingular elliptic curves. The Kummer surface is singular; the construction of S is as a minimal resolution. This is a result of Arthur Ogus.^[5] An extension to p = 2 has been made with a group scheme quotient.^[6]

[edit] Notes

^ In the case of a base field other than the complex numbers, the Betti number is that defined by the étale cohomology; the coefficients are l-adic numbers, with l some prime number different from the characteristic.
^ PDF, for characteristic 2, and Mathematische Annalen, Volume 328, Number 3, March 2004, pp. 451-468(18).
^ A N Rudakov, I R Šafarevič, "Supersingular K3 surfaces over fields of characteristic 2", Math. USSR Izv., 1979, 13 (1) 147-165.
^ Announcement [1].
^ ] Arthur Ogus, Supersingular K3 crystals, Journees de Geometrie Algebrique de Rennes(Rennes, 1978), Vol. II, Asterisque, vol. 64, Soc. Math. France, Paris, 1979, pp. 3–86.
^ PDF