Agnes Scott College

Marian Marsh Torrey

Classification of Monoidal Involutions Having a Fixed Tangent Cone



If a (1, 2) algebraic correspondence is established between the points of two spaces (x') and (x), such that a point P1 of (x) uniquely determines a point P' of (x'), but the point P' determines two points of (x), P1 and a second point P2, P1 and P2 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 P1 and P2 coincide. The locus of such points P' is called the surface of branch points L'. its image, the locus of the point P1- P2, is called the surface of coincident points, K. This latter surface is always a component of the Jacobian of the web |Sn|, 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.