This article focuses on a
certain class of surfaces in P5, exploiting the interplay between local projective
differential geometry and global algebraic geometry. These are the so-called
hypo-osculating surfaces, which have the property that at every point there is a
hyperplane which is doubly tangent. The first main result of the paper, obtained by
applying E. Cartan’s method of moving frames, is that locally any such
surface arises as a vector solution of one of the classical partial differential
equations of physics (the wave equation or a generalized heat equation).
Such equations were studied from a similar vantage point by G. Darboux
and C. Segre, among others. The remainder of the paper is concerned with
the global classification problem. Standard techniques in singularity theory
yield formulas for the inflection cycles, and in the case of ruled surfaces
there is an explicit numerical formula. By combining the earlier differential
geometric results with Kodaira’s classification of surfaces, one is able to
arrive at a fairly complete understanding of the inflectionary behavior of
hypo-osculating surfaces. In particular, the embedding of P1× P1 as the
quartic scroll is conjecturally the unique such smooth surface which is totally
uninflected.
