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On the quintic surface in space of five dimensions. (Part II, Curves on the surface.*)

Published online by Cambridge University Press:  24 October 2008

F. Bath
F. Bath
Affiliation:
King's College
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Extract

Consider the pencil of [4]'s through the [3] which contains l15, l25, l34, any [4] of the pencil meets the V25 again in a conic, q.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 2 , April 1928 , pp. 191 - 209
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

† V 25 denotes the quintic surface (or twofold).Google Scholar

* These results may be immediately verified from the parametric equations given in §4, since the functions giving the coordinates of a point on the surface are quadratic in each λ.Google Scholar

* This may also be seen if the representation be taken as obtained by projecting the surface into π from the plane containing l 14 and l 23.Google Scholar

† It should be noted that a point common to two curves in π, corresponds to a point common to the corresponding curves of V 25except in the case of the four fundamental points. If two curves pass through A 16, since A 16 corresponds to the line l 15, the corresponding curves on the surface each meet l 15 but not in general at the same point.Google Scholar

‡ Cf. Figure in Part I of this paper.Google Scholar

* These, and other intersection properties given in the sequel, are found from the plane representation.Google Scholar

* Cf. Baker, H. F., Principles of Geometry, Vol. 3, p. 189Google Scholar, and Proc. London Math. Soc. (2) 11 (1912), p. 285.Google Scholar

† Other than those lying in a [4].Google Scholar

* Castelnuovo, : Torino Atti, 24 (1889), p. 346.Google Scholar The maximum genus of a curve of order n in [r] is

where χ is an integer such that

† Thus any curve (2; 1, 0, 0, 0) is residual with any curve (4; 1, 2, 2, 2) and any curve (3; 2, 1, 1, 0) is residual with any curve (3; 0, 1, 1, 2).Google Scholar

‡ Both of these sets give the family (2; 1, 0, 0, 0), which is also given by the sets (l 14, l 23, l 15, l 34, l 12), (l 12, l 34, l 15, l 24, l 13, ) of the former type and (l 25, l 12, l 13, l 14, l 34), (l 35, l 12, l 24, l 14, l 13) of the latter type. The change from one of these sets to another of the same type is made by interchanging 2 and 3, or 3 and 4, or 4 and 2 in the suffixes.Google Scholar

* Cmp will denote a curve of order m and genus p.Google Scholar

* It is assumed that the Q 4 does not touch the V 25.Google Scholar

† A general canonical curve of genus six is one whose moduli are arbitrary; geometrically the distinction is that the curve has no trisecant line and no pentasecant plane (see § 8·1).Google Scholar

‡ Denoted in the sequel by C 106.Google Scholar

§ Cf. Enriques-Chisini, , Teoria Geometrica delle Equazioni, Vol. 3, p. 118.Google Scholar

* The general plane quintic is a well-known exception. It is of genus six, but Las upon it a g 52 and there is no such series on C 106 that lies on V 25.

For if C 106 has upon it a g 52, let i be its index of speciality; then 5 − 6 + i=2. Hence i = 3 and there are three linearly independent [4]'s through each set of the series. Each set, G 5, therefore lies in a plane. Now every quadric fourfold containing the curve is met by the plane of G 5 in a conic containing the five points of G 5; but there is only one conic through five points, therefore this conic, and all the conies similarly determined by g 52, lie upon all six linearly independent Q 4's containing C 106. V 25 contains no doubly infinite system of conies, nor does it lie on more than five linearly independent Q 4's, hence if C 106 has a g 52 upon it, it cannot He upon a V 25.

Enriques has shown (loc. cit. p. 106 and Rendiconti Bologna, 23 (1919), p. 80)Google Scholar that when C 106 has a g 52 upon it, the six linearly independent Q 4's containing the curve have, as their complete intersection, a Veronese surface. Generally he obtains the theorem:

A curve of genus p>4, not hyperelliptic, has for canonical curve a , belonging to a linear system of quadric (p − 2)-folds of dimension

and forming the complete intersection of (P+1) linearly independent Qp−2's of the system. The exceptions to this result arise when the contains a g31, the curve then lies upon a rational scroll of order (p−2) common to all the (P+1) quadrics; and, for p = 6, the case when C106 contains a g52 and lies upon a Veronese surface common to all the quadrics.

† Cf. the theorem quoted above from Enriques.Google Scholar

* Enriques finds one other type of primitive special series, namely: the ∞2g 51's obtained from the [3]'s of quadric fourfolds having a line of double points and containing the curve.Google Scholar

† I.e., each of the ten lines is a chord of C 106.Google Scholar

‡ Such a curve can be projected into a plane quintic. It has a double infinity of pentasecant planes (cut by [4]'s through the given one) and the complete intersection of the Q 4's containing it is a Veronese surface (see third footnote to § 8·1).Google Scholar

* We assume that no three of N 1, N 2, N 3, N 4 coincide; which is justified since, in that case, the corresponding three chords lie in a [3] with π and C 106 is met by this [3] in ten points. Similarly no two N's coincide, for, otherwise, C 106 has a g 21. We also assume that no three of the N's are collinear. When this assumption is not justified C 106 has only four g 41's and is special.Google Scholar

* The [3] cannot contain ω or π, because C 106 cannot have a ten secant [3].Google Scholar

* The [4] containing L, l 25, l 35 and the plane containing L, l 45 meet in one line l which passes through L and meets l 45; l, l 25, l 35 lie in the [4] and are met by a unique line l′. The plane (ll′) meets all four lines l 15. Clearly, if there is a second plane possessing this property, then l 25, l 35, l 45 lie in the [4] and N 2, N 3, N 4 will be collinear. This is not the case for the general C 105 we have taken. (Cf. note to § 8·430.)Google Scholar

* As before lik = lki. In the diagram, the intersections of the lines lik and the conics c 5π, c 5ω among themselves are marked with dots; their intersections with C 106 are marked with crosses. The skew hexagon of the diagram in § 1 corresponds to the hexagon here shown by the heavy line.Google Scholar

* Cf. the diagram of †8·462.Google Scholar

† See § 4·3.Google Scholar

‡ Cf. Enriques, quoted in note to § 8·1.Google Scholar