Adam Coffman

Real congruence of complex matrix pencils and complex projections of real Veronese varieties

Quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the Veronese embedding. A classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils. The images of some low-dimensional cases include certain quartic curves in the Riemann sphere, models of the real projective plane in complex projective 4-space, and some normal form varieties for real submanifolds of complex space with CR singularities.

Smooth counterexamples to strong unique continuation for a Beltrami system in C 2

We construct an example of a smooth map ℂ → ℂ2 which vanishes to infinite order at the origin, and such that the ratio of the norm of the derivative to the norm of the z derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.

Weighted Projective Spaces and a Generalization of Eves' Theorem

For a certain class of configurations of points in space, Eves’ Theorem gives a ratio of products of distances that is invariant under projective transformations, generalizing the cross-ratio for four points on a line. We give a generalization of Eves’ theorem, which applies to a larger class of configurations and gives an invariant with values in a weighted projective space. We also show how the complex version of the invariant can be determined from classically known ratios of products of determinants, while the real version of the invariant can distinguish between configurations that the classical invariants cannot.

REAL EQUIVALENCE OF COMPLEX MATRIX PENCILS AND COMPLEX PROJECTIONS OF REAL SEGRE VARIETIES

Quadratically parametrized maps from a product of real projective spaces to a com- plex projective space are constructed as the composition of the Segre embedding with a projection. A classification theorem relates equivalence classes of projections to equivalence classes of complex matrix pencils. One low-dimensional case is a family of maps whose images are ruled surfaces in the complex projective plane, some of which exhibit hyperbolic CR singularities. Another case is a set of maps whose images in complex projective 4-space are projections of the real Segre threefold.