† 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

* C_m^p 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 Q_p−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 l_ik = l_ki. In the diagram, the intersections of the lines l_ik and the conics c ₅^π, c ₅^ω 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