1996 Report Curvature of Algebraic Singularities

Curvature of Algebraic Singularities (Led by Donal O'Shea).

O'Shea's group consisted of Tom Cecil (Notre Dame), Albert Chau (Queen's), Wei Ding (MHC), Kaj Gartz (Yale), Craig Grabowski (Canisius), Yihua Jing (MHC), and Tom Wieand (Boston U). The group continued and extended work begun by Alan Durfee and Don O'Shea's group in the summer of 1994 studying real tangent cones to algebraic surfaces.

Recall that if V is a real surface in three-space given by f(x) = 0 , where f is a polynomial, and if f(0) = 0, the real (geometric) tangent cone to V at 0 is the set C(V, 0) of all vectors in three-space which lie on any line which can be obtained as a limit of secant lines one of whose endpoints goes through the origin and whose other endpoint tends to the origin. If the origin is a singular point, it turns out to be very difficult to compute the tangent cone. In particular, unlike the situation over the complexes, it is not given by the vanishing of the lowest degree homogeneous part of the polynomial f.

The students carefully analysed the work of the 1994 group, finding alternate derivations of the real tangent cones of all singularities of the form xA+D + xD yB + zC = 0 , the "Trotman family". They were also able to compute the conormal cone for all members of this family and to show by brute force that no member of the Trotman family has more than 4 real exceptional lines. (The conormal cone consists of all vectors in three-space which lie on any line which can be obtained as a limit of normal lines at smooth points of V which tend to the origin.) We still do not have a direct proof of this result.

Albert Chau and Tom Wieand also proved that any angular neighborhood of a real exceptional line contains arcs along which the principle curvatures of the surface go to infinity. This result had been conjectured some years ago, but the weak assumptions under which the students proved it were a total surprise. The work has been written up for publication and submitted to one of the American Mathematical Society journals.

Craig Grabowski and Wei Ding were able to find ways to create surfaces with preassigned conormal contents. This, too, has been written up and will be submitted once it is in final form.