- Supersingular K3 surface
In
algebraic geometry , a supersingular K3 surface is a particular type ofK3 surface . Such analgebraic surface has itscohomology generated byalgebraic cycle s; in other words, since the secondBetti number [In the case of a base field other than the complex numbers, the Betti number is that defined by theétale cohomology ; the coefficients arel-adic number s, with "l" someprime number different from the characteristic.] of a K3 surface is always 22, such a surface must possess 22 independent elements in itsPicard 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 theHodge 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).History
These surfaces were first discovered by
André Weil andJohn Tate and then more fully developed byMichael Artin andTetsuji Shioda .It has been conjectured, by Artin, that every supersingular K3 surface isunirational ; this conjecture remains openas of 2007 . Shioda has shown that supersingular K3 surfaces aredouble cover s of theprojective plane . [ [http://www.intlpress.com/AJM/p/2004/8_3/AJM-8-3-531-586.pdf PDF] , for characteristic 2, and Mathematische Annalen, Volume 328, Number 3, March 2004, pp. 451-468(18).] In the case of characteristic 2 the double cover may need to be aninseparable covering .Artin's conjecture has been shown to be true in characteristic two by
Shafarevich and Rudakov [ A N Rudakov, I R Šafarevič, "Supersingular K3 surfaces over fields of characteristic 2", Math. USSR Izv., 1979, 13 (1) 147-165.] 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 theintersection form on the Picard group of a supersingular K3 surface is an even power:"p"2"e"
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.
Examples
In characteristic two,
:"z"2 = "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 minimalmodel of the function field of such a surface is a supersingular K3 surface.The largest Artin invariant here is ten.
Similarly, in characteristic three,
:"z"3 = "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 minimalmodel 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 [Announcement [http://adsabs.harvard.edu/abs/2006math.....11452P] .] that every supersingular K3 surface with Artin invariant less than four is birationally equivalent to a surface with equation
:"z"5 = "h"("x", "y") ,
and thus every such surface is unirational.
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 surface s, in other words quotients of anabelian surface "A" by the mapping "x" → − "x". More precisely, "A" should be a supersingular abelian surface, isogenous to a product of twosupersingular elliptic curve s. The Kummer surface is singular; the construction of "S" is as a minimal resolution. This is a result ofArthur Ogus . [] 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.] An extension to "p" = 2 has been made with agroup scheme quotient. [ [http://www.math.hokudai.ac.jp/~shimada/preprints/Kummer/Kummer.pdf PDF] ]Notes
External links
* [http://www.math.lsa.umich.edu/~idolga/leech.pdf Construction of a surface (PDF)] related to the
Leech lattice
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