Some Types of Irregular Surfaces

Abstract

It is a problem of considerable interest in the theory of surfaces to determine the irregular non-singular surface of minimum order, not referable to a scroll; in previous investigations the author has discussed the regularity or referability of surfaces in higher space, reaching the conclusion that all non-singular surfaces of order n ≤ 10 in S4 are regular or referable, with the possible exception of the surface of order n = 10 and sectional genus π = 6, which may be elliptic (pg = 0, pa = −1) or hyperelliptic (pg = 1, pa = −1). In their memoir on hyperelliptic surfaces, Enriques and Severi have obtained for the irregular hyperelliptic surface of general moduli a model 6F10 of minimum order, situated in S4, with the characters n = 10, π = 6. Using transcendental methods, Comessatti has constructed a class of irregular hyperelliptic surfaces the properties of which he has examined in detail; this class includes a member 6Π10 which is a special case of 6F10; and since Comessatti has shown that Π10 is without singularities, so also is F10, whence it follows that F10 is a solution of the proposed problem.