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Unformatted text preview: quartic surface in P3
T0 + T 1 + T 2 + T 3 + T 0 T1 + T 0 T2 + T 1 T2 + T 0 T1 T2 ( T0 + T1 + T 2 ) = 0 (see ). Note that over a ﬁeld of positive characteristic the Hodge structure and
the inequality ρ = rankSX ≤ 20 does not hold. However, one can show that for
any K3 surface over an algebraically closed ﬁeld of positive characteristic
(5.11) ρ ≤ 22, ρ = 21, K3 surfaces with ρ = 22 are called supersingular (in the sense of Shioda). Observe
the striking analogy of inequalities (5.11) with the inequalities from Theorem 4.7.
Besides scaling, one can consider the following operations over nondegenerate
quadratic lattices which preserve the reﬂectivity property (see ). The ﬁrst operation replaces a lattice M with p−1 (M ∩ p2 M ∗ ) + M for any p dividing the
discriminant of M . This allows one to replace M with a lattice such that the exponent of the discriminant groups is square free. The second operation replaces
M with N (p), where N = M ∗ ∩ p−1 M . This allows one to replace M with a
lattice such that the largest power a of p dividing the discriminant of M satisﬁes
a ≤ 1 rankM .
I conjecture that up to scaling and the above two operations any even hyperbolic
reﬂective lattice is isomorphic to the lattice SX (−1) for some K3 surface deﬁned
over an algebraically closed ﬁeld of characteristic p ≥ 0.
It is known that the lattice SX (−1) for a supersingular K3-surface X over a ﬁeld
of characteristic p > 0 is always of rank 22 and its discriminant group is isomorphic
to a p-elementary group (Z/pZ)2σ , σ ≤ 10 (see ). No two such lattices are
equivalent in the sense of the operations on lattices described above. There is only
one such reﬂective lattice, namely U ⊥ D20 (−1).
There are only a few cases where one can compute explicitly the automorphism
group of a K3 surface when it is inﬁnite and the rank of SX is large. This requires
one to construct explicitly a Coxeter polytope of Ref2 (SX (−1)) which is of inﬁnite
volume. As far as I know this has been accomplished only in the following cases:
• X is the Kummer surface of the Jacobian variety of a general curve of genus
2 (); 34 IGOR V. DOLGACHEV • X is the Kummer surface of the product of two nonisogeneous elliptic curves
• X is birationally isomorphic to the Hessian surface of a general cubic surface
• SX (−1) = U ⊥ E8 ⊥ A2 ();
• SX (−1) = U ⊥ E8 ⊥ A2 ();
• SX is of rank 20 with discriminant 7 ();
• SX (−1) = U ⊥ D20 () (characteristic 2).
What is common about these examples is that the lattice SX (−1) can be primitively embedded in the lattice II25,1 from Example 4.6 as an orthogonal sublattice
to a ﬁnite root sublattice of II25,1 . We refer to  for the most general method
for describing Aut(X ) in this case.
5.4. Enriques surfaces. These are the surfaces from Case 2 (ii). They satisfy
2c1 (X ) = 0, c1 (X ) = 0, H1 (X, Z) = To...
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