REU 1995: Curvature of algebraic singularities (A. Durfee)

1. Curvature near singular points of algebraic curves and surfaces, both real and complex. (This project continued the 1988 and 1990 REU projects of O'Shea, and 1994 REU project of Durfee and O'Shea. It was followed by the 1996 REU project of O'Shea.) Let f(x₁, x₂, ..., x_n) be a real or complex polynomial which vanishes at the origin. If the partial derivatives vanish at the origin then this point is a singular point of the hypersurface defined by the zero locus of the polynomial in n-space. One can investigate the curvature of this hypersurface near the origin, and also the curvature of nearby smooth hypersurfaces. In particular one can look at the limit of the Gaussian and principal curvatures along paths in n-space ending at the origin. This group investigated this problem in the cases n=2 and n=3. In particular, they investigated whether these limiting curvatures depended just on the tangent vector of the path at the origin and not the path itself. (See the papers by Hemmer, Starr and Tsai below.)
2. Polynomial realizations of knot types (See the paper by Auerbach below.)
3. Total curvature of algebraic knots (See the paper by Brutt below.)

Student participants

• Ruth Auerbach, Haverford College '96
• LeeAnne Brutt, Allegheny College '96
• David Hemmer, Dartmouth College '96
• Jason Starr, University of California, Berkeley '96
• Harrison Tsai, Harvard College '96

Reports

David Hemmer, Limiting curvature near singular points of algebraic curves

Abstract: This paper examines the limiting curvature of algebraic plane curves near a singular point p. The limiting curvature is shown to be determined completely by the limiting tangent directions of the curve at p except for at finite number of ``bad'' directions, which are completely classified. The results easily extend to limiting Gaussian curvatures of complex algebraic curves. Using these results, a similar analysis is performed on limiting Gaussian curvature of real surfaces of the form f(x,y) + zⁿ = 0 near (0,0,0). This analysis permits several results to be obtained about limiting curvatures of Brieskorn singularities x^a+y^b+z^c=0. Finally, for these examples, a conjecture is given relating these bad directions to the exceptional lines of the surface.

Jason Starr and Harrison Tsai, Rational functions on algebraic varieties

Abstract: Let f(x,y,z) be a real or complex polynomial with f(0,0,0)= 0, and let T be a rational function on three space which is possibly undefined at the origin (for example, the curvature of the level surfaces of the above polynomial). This paper investigates the limiting behavior of T along paths lying in the surface f(x,y,z)=0 which end at the origin. A tangent direction at the origin is called ``good'' if the limiting behavior of T is independent of the path chosen, and ``bad'' otherwise. An algorithm is presented for finding the bad directions.

Ruth Auerbach, Loose-ended knots

Abstract: A (polynomial) loose-ended knot is a (polynomial) embedding of an interval into the three-ball whose endpoints, and only these points, lie on the boundary of the ball. It is shown that loose-ended knots are the same as usual knots. The paper contains two (unpublished) results due to Lee Rudolph relating the degree of a polynomial loose-ended knot to standard knot invariants. Polynomial representatives of the (5,2) and (7,2) torus knot are given.

LeeAnne Brutt, Total curvature of curves parameterized by polynomials

Abstract: The total curvature of a polynomial embedding of the line into three-space is related to the degree of the polynomial using a result of Milnor, and a conjecture is given.