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Duality in Projective Geometry

Dana S. Scott
Organization: Carnegie Mellon University
Department: Computer Science Department
URL: http://www-2.cs.cmu.edu/~scott/

2006 Wolfram Technology Conference
Conference location

Champaign IL

On four years (1989, 1993, 1997, 1998), the author offered a one-semester course on algebraic curves in the classical complex projective plane. The well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often affected by conic sections). What is not so well known is that this duality can be extended to higher-degree loci, relating point ranges (i.e., curves) to line configurations (i.e., envelopes of tangents). In the plane over the complex numbers, a homogeneous polynomial a of degree a (representing the point locus a) can be related to an envelope of degree a, say a, written in line coordinates, by regarding the symbols a, a, and a as the partial differential operators with respect to a, a, and a. In fact, the relationship is completely symmetric in that we can as well regard the symbols a, a, and a as the partial differential operators with respect to a, a, and a. Combining algebra with partial differentiation allows for symbolic proofs of properties of curves with simple conditions for representing singular points or tangents. These proofs can be materially aided with a judicious use of computer algebra. We recall that the linear homogeneous a-polynomials, a, can be regarded as lines; while the linear a-polynomials, a, can be taken as points (up to a constant, nonzero factor, as with homogeneous coordinates). Write a for, say, letting a differentiate a. (In this linear case, partial differentiation works just like a dot product between vectors.) Then the equation a means that the point a lies on the line a. The equation also means the line a passes through the point a, and, hence, is the basis for the duality between points and lines. What about higher degrees? We need a special case of a theorem of Euler, namely, if a is a homogeneous a-polynomial of degree a, and if a is the linear form a, then we have a, where a is shorthand for the evaluation of the polynomial as a function of a at a. Among other consequences, this theorem tells us that the equation a means that the point a lies on the curve a. But there are many algebraic consequences. For example, by a simple use of the binomial theorem, we can argue that if a represents an a-tuple point of the curve a, then the polynomial a of degree a factors into the a linear factors representing the tangents to the curve a at the point a. Mathematica well implements commutative algebra (and many algebraic algorithms) and of course it implements partial differentiation. It does not directly implement an algebra of differential operators, however. The author—with the help of graduate-student assistants—was able to do that by transcribing an axiomatic characterization of a into rewrite rules between the two kinds of polynomials using Mathematica rule sets. One big advantage discovered by using symbolic computation in this way was that formulas for the solution of geometric problems could be developed and then used in numerical computation (for example, in creating graphical illustrations). The improvement in Mathematica’s a routine made good pictures of (real parts of) curves possible. Over the years of course development, the author found this approach to classical, plane projective geometry most satisfactory, as the symbolic polynomial algebra really gives a middle way between synthetic and analytic geometry. The point is that symbolic computation allows one to name geometric objects by symbols and then to apply only those rules appropriate to the problem at hand. The trick is to find the right rules! Still, the experience in this course development was encouraging in that the algebra generated could not be carried out with any pleasure by hand, and the use of the computer definitely helped the author to think. But, more developments are needed before courseware can be written that fully uses theorem proving. Despite this success, several difficult problems remain: (1) How to extend this approach to fields of characteristic p? (2) How to use symbolic computation with algebraic numbers? (3) How to refine the methods to apply better to the real projective plane? (4) How to move to higher dimensions beyond plane geometry? (5) How to implement more of the ideas of invariant theory? (6) How to use symbolic computation in ideal theory? (7) How to connect the geometry with automated theorem proving? In regards to point (7), the author recently remarked that the multi-sorted first-order theory of the structure a is decidable. Here a is the space of homogeneous a-polynomials of degree a over the complex field a, and a is the space of homogeneous a-polynomials of degree a. This observation perhaps gives hope to a project of augmenting computer algebra by computer-assisted theorem proving — at least in geometry.

*Mathematics > Algebra > Polynomials
*Mathematics > Calculus and Analysis > Differential Geometry

algebraic curves, duality, differential operators, polynomial
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Duality.nb (1.3 MB) - Mathematica Notebook [for Mathematica 5.2]