### Classification of Monoidal Involutions Having a Fixed Tangent Cone

Introduction

If a (1, 2) algebraic correspondence is established between the points
of two spaces (x') and (x), such that a point P_{1} of (x) uniquely determines
a point P' of (x'), but the point P' determines two points of (x), P_{1} and a
second point P_{2}, P_{1} and P_{2} are conjugate points in a rational involution.
The correspondence between the spaces is defined by associating with
the planes of (x') a web of surfaces, order n, of (x), such that any three
surfaces of the web, not all belonging to the same pencil, intersect in only
two variable points. These two points correspond to the one point in (x')
which is the intersection of the three associated planes, and they are two
conjugate points of the involution. For certain points P' it happens that
the images P_{1} and P_{2} coincide. The locus of such points P' is called the
surface of branch points L'. its image, the locus of the point P_{1}- P_{2}, is
called the surface of coincident points, K. This latter surface is always a
component of the Jacobian of the web |S_{n}|, images of the planes of (x').

The first systematic discussion of (1, 2) correspondence between two
spaces was given by De Paolis. Previously De Paolis had studied the (1, 2)
correspondence between two planes, as had Noether and others. This space
transformation has proved useful in the investigation of the singularities of
certain surfaces, for example, the Kummer surface, and has also been used
in the study of space involutions by Sharpe and Snyder. In the present
paper it is used in discussing some types of monoidal involutions in which
the monoidal transformation is compounded with the homology, the Geiser,
the Bertini and the Jonquieres involutions in the bundle. In addition all
monoids of the web defining the (1, 2) correspondence have a fixed tangent
cone.