The Picard number of a Kummer K3 surface
Let be a separably closed field of characteristic not , and an abelian surface. Then it is a basic fact (e.g. see Example 1.3 (iii) of Huybrechts’ “K3 Surfaces”) that one can make a K3 surface out of . The construction is as follows. Consider the involution given by The fixed locus of this involution is exactly , a finite constant closed -subgroup scheme of with -points. Let , and consider the blow-up . By the universal property of the blow-up, the involution lifts to a map such that the diagram
commutes. Furthermore, by the uniqueness part of the statement of the universal property of , we deduce easily that is also an involution.
Now the blow-up is a projective variety over with an action of via the involution . Therefore, the categorical quotient exists in the category of schemes. The scheme constructed in this way is called the Kummer surface associated to , and turns out to be a K3 surface. In particular, , so the Picard scheme of is étale, and is therefore a finitely-generated abelian group.
For an arbitrary proper scheme , recall that the Néron–Severi group of is a finitely generated abelian group (SGA 6, Exposé XIII, Théorème 5.1). Therefore, we may consider the Picard number , defined as the rank of the Néron–Severi group of (which for is just the rank of the Picard group). It is proven in Shioda’s paper “Supersingular K3 surfaces” that the Picard number of is given by the formula . However, we find his calculation unclear and difficult to follow. In this note, we give an explicit proof of this fact that is entirely self-contained. We do not use any Hodge theory, e.g. we do not study the complement of in , i.e. the transcendental lattice .
Main Theorem: Let be a separably closed field of characteristic not , and an abelian surface. Let denote the Kummer surface associated to . Then the Picard number of is given by
In this section, we record a crucial result about abelian varieties that we will need. Let be a separably closed field of characteristic not , and let be an abelian variety (of arbitrary dimension). The group acts on by , and this action descends to one on the subgroup of numerically trivial line bundles . In particular, the sequence
Proposition 1: Taking -invariants in (4), we obtain an exact sequence
where is the dual abelian variety of . In particular, is a finitely generated abelian group of rank equal to the Picard number of .
Let us first recall several facts about multiplication by on :
- For , (See the proof of Lemma 5.2.5 in Brian Conrad’s abelian varieties notes).
- For , we have
In other words, has the effect of multiplication by on (Lemma 7.5.2 of loc. cit.).
Granting these facts, let us first compute . By (I), a line bundle satisfies precisely when . In other words,
Next, by (II) above, the -action on is trivial, so . Finally, we show that , which will complete the proof of the proposition. Write for the generator of , and let denote the “norm” map
By the calculation of the cohomology of finite cyclic groups,
By (I) above, we have . On the other hand, for ,
In other words, has the effect of multiplication by on . But now recall that , and is surjective étale. Since is separably closed, the map on -points is surjective, and therefore , from which the vanishing of the cohomology group in question follows.
Proof of Main Theorem
Let be the points in , let be the preimage of in (the exceptional divisors), and let denote the image of in the quotient . Define
It is proven in Theorem 10.6 of Badescu’s “Algebraic Surfaces” that the ’s are irreducible divisors in , and are furthermore -linearly independent in . Therefore, identifying the Weil class group of with its Picard group (by smoothness), we obtain the exact sequence
Now the formation of the quotient commutes with open immersions. More precisely, for any open subscheme , if we let , then the map is the categorical quotient of by . Therefore,
But now observe that
Therefore, the result will follow from (5) if we can show that the abelian group is finitely generated of rank equal to . To this end, define and . Since the action of on is free, the quotient map is a Galois cover (SGA3, Exposé V, Théorème 4.1(iii) and (iv)), and we have an associated Hochschild-Serre spectral sequence
The low-degree terms of this spectral sequence are
Now before we calculate any cohomology, we make the observation that
Indeed, this is true by Hartogs’ Lemma since is smooth (a fortiori normal!) and is codimension in . Also, observe that the Galois action of on is trivial.
We now compute . By the discussion above, this is isomorphic to On the other hand, by the calculation of the cohomology of finite cyclic groups, . Since is separably closed, this is zero and so sits in an exact sequence6) yields the equality