The Elliptic Curve as Intersection of Two Quadrics in P 3

In this section we derive the embedding ofE with a zero pointPand the rational pointsQ, R andS into _P3 as the intersection of two non-generic quadrics. We follow the methods described in chapter 3 and in [97, 34] used for the derivation of the general elliptic curves with rank one and two Mordell-Weil groups.

We note that the presence of the four points on E defines a degree four line bundle O(P+Q+R+S)over E. Let us first consider a general degree four line bundleMover E. Then the following holds, as we see by employing the Riemann-Roch theorem:

1. H0(E,M)is generated by four sections, that we denote byu0,v0,w0,t0.

2. H0(E,M2_{)is generated by eight sections. However we know ten sections ofM}2_{, the} quadratic monomials in[u0:v0:w0:t0], i.e.u02,v02,w02,t02,u0v0,u0w0, u0t0, v0w0, v0t0, w0t0.

The above first bullet point shows that [u0:v0:w0:t0]are of equal weight one and can be viewed as homogeneous coordinates on_P3. The second bullet point implies thatH0(2M)

is generated by sections we already know and that there have to be two relations between the ten quadratic monomials in[u0:v0:w0:t0], that we write as

s₁t02+s₂u02+s₃v02+s₄w02+s₅t0u0+s₆u0v0+s₇u0w0+s₈v0w0=s₉v0t0+s₁₀w0t0, (5.1) s₁₁t02+s₁₂u02+s₁₃v02+s₁₄w02+s₁₅u0t0+s₁₆u0v0+s₁₇u0w0+s₁₈v0w0=s₁₉v0t0+s₂₀w0t0,

By insertingu0=0 into (5.1) we should then get four rational solutions corresponding to the four points, i.e. other words (5.1) should factorize accordingly. However, this is not true for genericsitaking values e.g. in the ring of functions of the baseBof an elliptic fibration1

Thus, we have to set the following coefficientssito zero,

s₁=s₃=s₄=s₁₁=s₁₃=s₁₄=0. (5.3) As we see below in section 5.1.2, this can be achieved globally, by blowing up_P3at three generic points.

For the moment, let us assume that (5.3) holds and determineP,Q,R,S. First we note that the presentation (5.1) for the elliptic curveE now reads

s2u02+s5u0t0+s6u0v0+s7u0w0 = s9v0t0+s10w0t0−s8v0w0, (5.4) s₁₂u02+s₁₅u0t0+s₁₆u0v0+s₁₇u0w0 = s₁₉v0t0+s₂₀w0t0−s₁₈v0w0,

which is an intersection of twonon-genericquadrics in_P3. Settingu0=0 we obtain

0=s₉v0t0+s₁₀w0t0−s₈v0w0, 0=s₁₉v0t0+s₂₀w0t0−s₁₈v0w0, (5.5)

1_{In contrast, if we were considering an elliptic curve over an algebraically closed field, we could set some}

si=0 by using thePGL(4)symmetries ofP3to eliminate some coefficientssi. For example,s3=0 can be achieved by making the transformation

u07→u0+kv0, withkobeying (s2k2+s6k+s3) =0. (5.2) Solving this quadratic equation inkwill, however, involve the square roots ofsi, which is only defined in an algebraically closed field. In particular, when considering elliptic fibrations the coefficientssi will be represented by polynomials, of which a square root is not defined globally.

which has in the coordinates[u0:v0:w0:t0]the four solutions

P= [0 : 0 : 0 : 1], Q= [0 : 1 : 0 : 0], R= [0 : 0 : 1 : 0],

S= [0 :|M₁S||M₃S|:−|M₁S||M₂S|:−|M₃S||M₂S|]. (5.6) Here we introduced the determinants|M_iS|of all three 2×2-minorsM_iSreading

|M₁S|=s₉s₂₀−s₁₀s₁₉, |M₂S|=s₈s₁₉−s₉s₁₈, |M₃S|=s₈s₂₀−s₁₀s₁₈, (5.7) that are obtained by deleting the(4−i)-th column in the matrix

MS=    s₉ s₁₀ −s₈ s₁₉ s₂₀ −s₁₈   , (5.8)

whereMSis the matrix of coefficients in (5.5).

It is important to realize that the coordinates of the rational point S are products of determinants in (5.7), in particular when studying elliptic fibrations at higher codimension in the base B, cf. section 5.3. On the one hand, the vanishing loci of the determinant of a single determinant |M_iS| with i=1,2,3 indicates the collisions of S with P, Q and R, respectively, i.e.

|M₁S|=0 : S=P, |M₂S|=0 : S=Q, |M₃S|=0 : S=R. (5.9) On the other hand the simultaneous vanishing of all|M_iS|is equivalent to the two constraints in (5.4) getting linearly dependent. Then, the elliptic curve E degenerates to an I₂-curve, i.e. two_P1’s intersecting at two points, see the discussion around (5.27), with the pointS

becoming the entire_P1={u=s₉v0t0+s₁₀w0t0−s₈v0w0=0}2_{. We note that this behavior} ofSindicates that in an elliptic fibration the pointS will only give rise to a rational, not a holomorphic section of the fibration.

In summary, we have found that the general elliptic curve E with three rational points Q, R,S and a zero pointPis embedded into_P3 as the intersection of the two non-generic quadrics (5.4).