Let c = (
c
1
, . . . ,
c
r
) be a set of curves forming a minimum base on a surface, which, under a self-transformation,
T
, of the surface, transforms into a set
T
c expressible by the equivalences
T
c = Tc, where T is a square matrix of integers. Further, let the numbers of common points of pairs of the curves,
c
i
,
c
j
, be written as a symmetrical square matrix Г. Then the matrix T satisfies the equation TГT' = Г. The significance of solutions of this equation for a given matrix Г is discussed, and the following special surfaces are investigated: §§4-7. Surfaces, in particular quartic surfaces, wìth only two base curves. Self-transformations of these depend on the solutions of the Pell equation
u
2
-
kv
2
= 1 (or 4). §8. The quartic surface specialized only by being made to contain a twisted cubic curve. This surface has an involutory transformation determined by chords of the cubic, and has only one other rational curve on it, namely, the transform of the cubic. The appropriate Pell equation is
u
2
- 17
v
2
= 4. §9. The quartic surface specialized only by being made to contain a line and a rational curve of order
m
to which the line is (
m
- 1)⋅secant (for
m
= 1 the surface is made to contain two skew lines). The surface has two infinite sequences of self-transformations, expressible in terms of two transformations
R
and
S
, namely, a sequence of involutory transformations
R
S
n
, and a sequence of non-involutory transformations
S
n
.
Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds