Pencils of circles with a straight line and circle as the basic elements

Pencils of circles with are a straight line and a circle as the basic elements are investigated. Three cases of arrangement of a basic straight line and a circle are considered: when the straight line does not intersect a circle, when the straight line and a circle have one generic point, and when the straight line intersects a circle in two points. A parameter is entered and the equations of new pencils of circles are registered. By means of mathematical manipulations the obtained equations are given to the initial equation of a circle. Different values are attached to the parameter and the circles belonging to new pencils are constructed. Based on the obtained graphs it is concluded that the pencil with not intersecting basic straight line and a circle forms a hyperbolic pencil of circles, a pencil with a basic straight line and a circle having one generic point forms a parabolic pencil, and a pencil with the intersecting basic straight line and a circle forms an elliptic pencil.

Pseudo-Abelian varieties

It is a familiar fact that the Picard surface (or hyperelliptic surface of rank 1) admits a completely transitive permutable continuous group of ∞2 automorphisms. There are, however, other non-scrollar surfaces which possess continuous groups of automorphisms, namely, the elhptic surfaces. Every elliptic surface V2 contains a pencil of birationally equivalent elhptic curves, which are the trajectories of the group in question; it also contains a second, elliptic, pencil of birationally equivalent curves; the intersection number of the two pencils is an important character, known as the determinant of V2. Just as any Picard surface can be mapped on a multiple Picard surface of divisor unity, so V2 can be mapped on a multiple elliptic surface of determinant unity, the branch curve (if any) corresponding to a certain number of trajectories.