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 [154, 156] 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)overE. Let us first consider a general degree four line bundleMoverE. Then the following holds, as we see by employing the Riemann-Roch theorem:

2. H0(E,M2)is generated by eight sections. However we know ten sections ofM2, the quadratic monomials in[u′∶v′∶w′∶t′], i.e.u′2,v′2,w′2,t′2,u′v′,u′w′,u′t′,v′w′,v′t′, w′t′.

The above first bullet point shows that[u′∶v′∶w′∶t′] are of equal weight one and can be viewed as homogeneous coordinates on_P3_{. The second bullet point implies thatH}0_(2M)

is generated by sections we already know and that there have to be two relations between the ten quadratic monomials in[u′∶v′∶w′∶t′], that we write as

s₁t′2+s₂u′2+s₃v′2+s₄w′2+s₅t′u′+s₆u′v′+s₇u′w′+s₈v′w′=s₉v′t′+s₁₀w′t′, (2.1) s₁₁t′2+s₁₂u′2+s₁₃v′2+s₁₄w′2+s₁₅u′t′+s₁₆u′v′+s₁₇u′w′+s₁₈v′w′=s₁₉v′t′+s₂₀w′t′,

Now specialize toM = O(P+Q+R+S)and assumeu′ to vanish at all pointsP,Q,R,S. By insertingu′=0 into (2.1) we should then get four rational solutions corresponding to the four points, i.e. other words (2.1) should factorize accordingly. However, this is not true for generics_i taking values e.g. in the ring of functions of the base Bof an elliptic fibration6

Thus, we have to set the following coefficientss_ito zero,

s₁=s₃=s₄=s₁₁=s₁₃=s₁₄=0. (2.3)

As we see below in section 2.2.2, this can be achieved globally, by blowing up_P3_{at three}

generic points.

For the moment, let us assume that (2.3) holds and determine P,Q,R,S. First we note 6_{In contrast, if we were considering an elliptic curve over an algebraically closed field, we could set some} si=0 by using the_PGL(4)symmetries of_P3to eliminate some coefficientssi. For example,s3=0 can be achieved by making the transformation

u′

↦u′+kv′, withkobeying (s2k2+s6k+s3) =0. (2.2)

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.

that the presentation (2.1) for the elliptic curveE now reads

s₂u′2+s₅u′t′+s₆u′v′+s7u′w′ = s9v′t′+s10w′t′−s8v′w′, (2.4)

s₁₂u′2+s₁₅u′t′+s₁₆u′v′+s₁₇u′w′ = s₁₉v′t′+s₂₀w′t′−s₁₈v′w′,

which is an intersection of twonon-genericquadrics in_P3_{. Settingu}′_{=0 we obtain}

0=s₉v′t′+s₁₀w′t′−s₈v′w′, 0=s₁₉v′t′+s₂₀w′t′−s₁₈v′w′, (2.5)

which has in the coordinates[u′∶v′∶w′∶t′]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∣]. (2.6)

Here we introduced the determinants∣M_iS∣of all three 2×2-minorsM_iSreading

∣M₁S∣ =s9s20−s10s19, ∣M₂S∣ =s8s19−s9s18, ∣M₃S∣ =s8s20−s10s18, (2.7)

that are obtained by deleting the(4−i)-th column in the matrix

MS= ⎛ ⎜ ⎜ ⎝ s₉ s₁₀ −s₈ s19 s20 −s18 ⎞ ⎟ ⎟ ⎠ , (2.8)

whereMS_{is the matrix of coefficients in (2.5).}

It is important to realize that the coordinates of the rational point S are products of determinants in (2.7), in particular when studying elliptic fibrations at higher codimension in the base B, cf. section 2.4. On the one hand, the vanishing loci of the determinant of

respectively, i.e.

∣M₁S∣ =0∶ S=P, ∣M₂S∣ =0∶ S=Q, ∣M₃S∣ =0∶ S=R. (2.9)

On the other hand the simultaneous vanishing of all∣M_iS∣is equivalent to the two constraints in (2.4) getting linearly dependent. Then, the elliptic curve E degenerates to an I₂-curve, i.e. twoP1’s intersecting at two points, see the discussion around (2.27), with the pointS becoming the entire_P1_{= {u=s}

9v′t′+s10w′t′−s8v′w′=0}7. We note that this behavior of S indicates that in an elliptic fibration the point S will only give rise to a rational, not a holomorphic section of the fibration.

In summary, we have found that the general elliptic curveE with three rational points Q, R,S and a zero pointPis embedded into_P3 as the intersection of the two non-generic quadrics (2.4).