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Supersingular K3 surface
In algebraic geometry, a supersingular K3 surface is a K3 surface over a field k of characteristic p > 0 such that the slopes of Frobenius on the crystalline cohomology H2(X,W(k)) are all equal to 1.[1] These have also been called Artin supersingular K3 surfaces. Supersingular K3 surfaces can be considered the most special and interesting of all K3 surfaces.
Definitions and main results
More generally, a smooth projective variety X over a field of characteristic p > 0 is called supersingular if all slopes of Frobenius on the crystalline cohomology Ha(X,W(k)) are equal to a/2, for all a. In particular, this gives the standard notion of a supersingular abelian variety. For a variety X over a finite field Fq, it is equivalent to say that the eigenvalues of Frobenius on the l-adic cohomology Ha(X,Ql) are equal to qa/2 times roots of unity. It follows that any variety in positive characteristic whose l-adic cohomology is generated by algebraic cycles is supersingular.
A K3 surface whose l-adic cohomology is generated by algebraic cycles is sometimes called a Shioda supersingular K3 surface. Since the second Betti number of a K3 surface is always 22, this property means that the surface has 22 independent elements in its Picard group (ρ = 22). From what we have said, a K3 surface with Picard number 22 must be supersingular.
Conversely, the Tate conjecture would imply that every supersingular K3 surface over an algebraically closed field has Picard number 22. This is now known in every characteristic p except 2, since the Tate conjecture was proved for all K3 surfaces in characteristic p at least 3 by Nygaard-Ogus (1985), Maulik (2014), Charles (2013), and Madapusi Pera (2013).
To see that K3 surfaces with Picard number 22 exist only in positive characteristic, one can use Hodge theory to prove that the Picard number of a K3 surface in characteristic zero is at most 20. 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 h2,0 and h0,2 both take the value 1, with h1,1 = 20. Therefore the dimension of the space spanned by the algebraic cycles is at most 20 in characteristic zero; surfaces with this maximum value are sometimes called singular K3 surfaces.
Another phenomenon which can only occur in positive characteristic is that a K3 surface may be unirational. Michael Artin observed 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.[2] This is now known in every characteristic except 3. Artin's conjecture was proved in characteristic 2 by Rudakov & Shafarevich (1978), and in every characteristic p at least 5 by Liedtke (2013). Another proof for p at least 5 has been given by Lieblich (2014).
History
The first example of a K3 surface with Picard number 22 was given by Tate (1965), 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. Shimada (2004, 2004b) 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.
Kummer surfaces
If the characteristic p is greater than 2, Ogus (1979) showed that every K3 surface S with Picard number 22 and Artin invariant at most 2 is a Kummer surface, meaning the minimal resolution of the quotient of an abelian surface A by the mapping x ↦ − x. More precisely, A is a supersingular abelian surface, isogenous to the product of two supersingular elliptic curves.
See also
K3 surface
Tate conjecture
Notes
M. Artin and B. Mazur. Ann. Sci. École Normale Supérieure 10 (1977), 87-131. P. 90.
M. Artin. Ann. Sci. École Normale Supérieure 7 (1974), 543-567. P. 552.
References
Artin, Michael (1974), "Supersingular K3 surfaces", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 7: 543–567, MR 0371899
Charles, F. (2013), "The Tate conjecture for K3 surfaces over finite fields", Inventiones Mathematicae 194: 119–145, doi:10.1007/s00222-012-0443-y, MR 3103257
Lieblich, M. (2014), On the unirationality of supersingular K3 surfaces, arXiv:1403.3073
Liedtke, C. (2013), "Supersingular K3 surfaces are unirational", Inventiones Mathematicae, arXiv:1304.5623, doi:10.1007/s00222-014-0547-7
Madapusi Pera, K. (2013), "The Tate conjecture for K3 surfaces in odd characteristic", Inventiones Mathematicae, arXiv:1301.6326, Bibcode:2013arXiv1301.6326M, doi:10.1007/s00222-014-0557-5
Maulik, D. (2014), "Supersingular K3 surfaces for large primes", Duke Mathematical Journal 163: 2357–2425, arXiv:1203.2889, Bibcode:2012arXiv1203.2889M, doi:10.1215/00127094-2804783, MR 3265555
Nygaard, N.; Ogus, A. (1985), "Tate's conjecture for K3 surfaces of finite height", Annals of Mathematics 122: 461–507, doi:10.2307/1971327, MR 0819555
Ogus, Arthur (1979), "Supersingular K3 crystals", Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, Astérisque 64, Paris: Société Mathématique de France, pp. 3–86, MR 563467
Rudakov, A. N.; Shafarevich, Igor R. (1978), "Supersingular K3 surfaces over fields of characteristic 2", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 42 (4): 848–869, Bibcode:1979IzMat..13..147R, doi:10.1070/IM1979v013n01ABEH002016, MR 508830
Shimada, Ichiro (2004), "Supersingular K3 surfaces in characteristic 2 as double covers of a projective plane" (PDF), The Asian Journal of Mathematics 8 (3): 531–586, arXiv:math/0311073, Bibcode:2003math.....11073S, doi:10.4310/ajm.2004.v8.n3.a8, MR 2129248
Shimada, Ichiro (2004b), "Supersingular K3 surfaces in odd characteristic and sextic double planes", Mathematische Annalen 328 (3): 451–468, doi:10.1007/s00208-003-0494-x, MR 2036331
Shioda, Tetsuji (1979), "Supersingular K3 surfaces", Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732, Berlin, New York: Springer-Verlag, pp. 564–591, doi:10.1007/BFb0066664, MR 555718
Tate, John T. (1965), "Algebraic cycles and poles of zeta functions", Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), New York: Harper & Row, pp. 93–110, MR 0225778
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