# Algebraic Threefolds: With Special Regard to Problems of by Leonard Roth

By Leonard Roth

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Anotherresult due to COMESSATTI [1 J, also established by transcendental methods, is as follows: if the characters q2 (> 0), q3 of a threefold V3 satisfy the inequality q3 ~ q2 - 4, then Va contains a pencil of surfaces which, except when q2=1 (in which case the pencilis elliptic), has genus 2 at least. 1. Introduction. 39 Returning now to the case d = 2, we observe that an irregular non-scrollar surface for which pg = 0 necessarily has arithmetic genus Pa = -1 (ENRIQUES, a); moreover, it can be shown that, in addition to the irrational (elliptic) pencil which the surface contains, there is a second pencil, which is rational and free from base points.

6) The latter varies with Zi in a linear system which, by virtue of the generality of the forms /;, and the inequality (2), coincides with that of all primals V;=i of Sk' Now choose from the birational system of spaces Sh of Sk a system E of dimension k which depends rationally on k non-homogeneous parameters ~l' ~2' • • . '~k' It follows that a finite number fl of members of E lie on the generic V;=i of Sk' In order that a V~=~ of the linear system (6) should contain a space S h of E we must impose k linearly independent conditions on Zk+l' Zk+2' ...

A surface with rational curve sections; 2. a surface with elliptic curve sections and of order n ~ 2, which, for n > 2, is non-singular; 3. a surface with hyperelliptic curve sections, containing a rational pencil of conics; 4. a double quadric cone with sextic branch curve. It may be added that, if a simple model of 4. is required, we may take it to be a surface F9 mapped on the plane by novenic curves having 8 triple base points. , n = 2, is provided by a surface F8 mapped by sextics with 7 double base points.