By Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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Pr be the (infinitely near) base points of and m 1 , . . , m r the corresponding multiplicities. We may assume that m 1 ≥ m 2 ≥ · · · . The starting point is Noether’s inequality m 1 + m 2 + m 3 > n. 8. If the corresponding points P1 , P2 , P3 are actual points of P2 , then by a linear change of coordinates we can assume that these are (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1). Composing φ with the standard quadratic transformation gives P2 which is given by a linear subsystem of |(2n − m 1 − a new map φ : P2 m 2 − m 3 )H |.

First note that a smooth cubic surface in P3 containing two noncoplanar rational curves is unirational. 36. Because C 1 and C2 do not lie in the same plane, their join (meaning the locus of points lying on lines joining points on C1 to points on C 2 ) must be all of P3 . This ensures that the map φ is dominant, and hence finite. Because C 1 and C 2 are rational (over k), we conclude that X is unirational (over k). Thus we must find two non-coplanar rational curves on our cubic surface. 5 Cubic hypersurfaces 27 not Eckardt points.

It is easy to see that this surface does not contain any disjoint pair of lines defined over Q (or even over R) but that it does contain the conjugate pair of disjoint lines parameterized as L i = (w, − i w, i ), where i for i = 1, 2 are the complex cube roots of 1 and we work in the affine chart v = 0. Setting w = t + 1 s we obtain conjugate representations for the lines as i (s, t) = (t, s, 0) + i (s, s − t, 1). The line joining them has a parametric representation with parameter λ: (t, s, 0) + λ(s, s − t, 1).

### Algebra Some Current Trends by Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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