As we have seen in the previous chapter, the Kummer surface associated to a Jacobian surface can be embedded as a quartic hypersurface into P3. It turns out that the defining equations for the Kummer threefold are far more complicated, but at least we can still describe them explicitly.
Proposition 4.1. Let K denote the Kummer variety of a Jacobian of di-mension g ≥ 2, with embedding κ = (κ1, . . . , κ2g) into P2g−1. Then the image of K underκ can be described as an intersection of quartics.
Proof. This proof was suggested to me by Tzanko Matev. Let Q denote the set of monic quadratic monomials in the κi and let d≤ m denote the dimension of the space they generate, where m = 2g2+1
is the cardinality of Q. Let S = {s1, . . . , sd} be a subset of Q that is linearly independent inQ(f0, . . . , f7)[q1, . . . , qm], where the elements of Qare ordered so that we have Q={q1, . . . , qm} withqi =si fori= 1, . . . , d.
Letαdenote the 2-uple embedding ofP2g−1intoPm−1such that ifP ∈J, then we have
αi(κ(P)) =qi(P) for all i∈ {1, . . . , m}.
Then there are m−d linear relations on the image of K = κ(J) under α. Now consider an embedding β : J ֒→ P4g−1 given by a basis of L(4Θ) whose firstdelements are equal tos1, . . . , sd. Then we have a commutative diagram
J
κ
β //P4g−1
γ
P2g−1 α //Pm−1
where γ is a rational map defined as follows: If z = (z1, . . . , z4g), then γ(z) = y, where yi = zi for i = 1, . . . , d and the other yi are determined by the linear relations of Q. By construction, we have that β(J) lies in the domain of γ and in fact
γ(β(J))∼=α(κ(J)).
But it follows from [6, Theorem 7.4.1] that the image ofJ underβis defined by an intersection of quadrics, which then must hold for γ(β(J)) as well, since γ has degree 1. As the pullback underα of γ(β(J)) is isomorphic to K, the result follows.
Hence it suffices to find a basis for the space of quartic relations onK to describeK. We first compute a lower bound on the dimension of this space.
n m(n) e(n) d(n)
1 4 4 4
2 10 10 10
3 20 20 20
4 35 34 34
Table 4.1: Dimensions in genus 2 n m(n) e(n) d(n)
1 8 8 8
2 36 36 35
3 120 112 ≤112 4 330 260 ≤260
Table 4.2: Dimensions in genus 3
For n≥1 let m(n) denote the number of monic monomials of degree n in κ1, . . . , κ2g and letd(n) denote the dimension of the space spanned by them.
Then we havem(n) = 2g+n−1n
. Moreover, lete(n) denote the dimension of the space of even functions inL(2nΘ). By [6, Corollary 4.7.7] this is equal to (2n)g/2 + 2g−1. Since a monomial of degree n in theκi induces an even function inL(2nΘ), we always have d(n)≤e(n).
In genus 2, the dimension count is given in Table 4.1. We know thatd(4) can be at moste(4) = 34, and indeed the space of quadratic relations in the κi is one-dimensional, spanned by the Kummer surface equation (3.11).
Now we return to the case of genus 3. Stubbs has found the following quadratic relation between theκi and shown that it is unique up to scalars:
R1 :κ1κ8−κ2κ7−κ3κ6−κ4κ5−2f5κ2κ4+f5κ23+2f6κ3κ4+3f7κ24= 0 (4.5) The dimensions for genus 3 are presented in Table 4.2. The existence and uniqueness of R1 implies that d(2) = 35, but since e(2) = 36, this means that there is an element ofL(4Θ) not coming from a quadratic monomial in theκi, which does not happen in genus 2. Accordingly, we can at this point only boundd(3) andd(4) from above. It follows that in genus 3 there must be at least 70 = 330−260 quartic relations on the Kummer variety. But 36 of these are multiples of the quadratic relationR1, so there must be at least 34 genuine quartic relations.
In [100, Chapter 5] Stubbs lists, in addition to R1, 26 quartic relations and conjectures that these 27 relations are independent and form a basis of the space of all relations on the Kummer variety. These are the relations that are at most quadratic inκ5, . . . , κ8. He was not able to prove either of these conjectures. Using current computing facilities we can verify the former conjecture quite easily, but because of our dimension counting argument, we know that the latter conjecture cannot be correct.
x y
xi 1 0
yi 0 1
fi −i 2
κi, i≤4 i−1 0 κi, i >4 i−9 2
Table 4.3: x- and y-weight
How can such relations be derived? We employ the technique used al-ready by Stubbs to find his relations to obtain a complete system of defin-ing equations. Because of the enormous size of the algebra involved in these computations, we cannot simply search for relations among all mono-mials. Instead we split the monomials into parts of equal x-weight and y-weight. These are homogeneous weights discussed in [100,§3.5] that were already used by Flynn in [40] in order to derive quadratic relations defining a Jacobian surface in P15. See Table 4.3, reproduced essentially from [100, Figure 3.4].
On monomials of equal x- andy-weight we can use linear algebra to find relations; we continue this process with increasing weights until we have found enough quartic relations to generate a space of dimension 70. The difficulty of this increases mostly with the y-weight, the x-weight is not so important.
Theorem 4.2. There are relations R2, . . . , R35 which can be downloaded from the author’s homepage [74] such that the space of relations of degree at most 4 in (κ1, . . . , κ8) is generated byR1, . . . R35 and the largest y-weight of the Ri is 8.
Proof. Using a computer algebra system, for instanceMagma, one can check that R1, . . . , R35 are indeed relations on K and that they are independent.
For the latter it suffices to check that the space
W ={R2, . . . , R35} ∪ {κiκjR1 : 1≤i≤j≤8}
has dimension equal to 70 for one example, say for F(X, Z) = Z8+X7Z.
For this example, we can also compute all quartic relations and check that the space they generate has dimension 70 and equalsW. It now follows that the space of all quartic relations has dimension exactly 70 in general and we are done.
Corollary 4.3. The relations on the Kummer threefold are generated by the relations R1, . . . , R35.
Proof. Combine Proposition 4.1 with Theorem 4.2.
Next we generalize a useful notion from Chapter 3, namely that of Kum-mer coordinates.
Definition 4.4. Let l be a field of characteristic different from 2 with al-gebraic closure ¯l and let x = (x1, . . . , x8) ∈A8¯l \ {(0, . . . ,0)}. Let K ⊂ P7l be the Kummer variety associated with the Jacobian J of a smooth pro-jective genus 3 curve defined over l. We say that x is a set of Kummer coordinates on K if the image of x inP¯7l lies on K. If P ∈J, then we say thatxis aset of Kummer coordinates for P if xrepresentsκ(P), that is, if κ(P) = (x1 : . . . :x8). The set of all sets of Kummer coordinates on K is defined by
KA:={(x1, . . . , x8)∈A8 :∃P ∈K such thatP = (x1:. . .:x8)}.