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
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
Preliminaries
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
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,
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
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,
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
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 sequence