(Note that the room/time is DIFFERENT from usual!!)

We study rational curves in projective space, most notably rational plane curves, through the syzygy matrix of the forms parametrizing them. A rational plane curve C of degree d can be parametrized by three forms f_1,f_2,f_3 of degree d in the polynomial ring k[x,y], and the syzygy matrix F of these forms is easier to handle and often reveals more information than the implicit equation of C. Our goals are to read information about the singularities of C solely from the matrix F, to set up a correspondence between the types of singularities on the one hand and the shapes of the syzygy matrices on the other hand, and to use this correspondence to stratify the space of rational plane curves of a given degree.

The constellation of singularities is also reflected in strictly numerical information about the Rees ring of the ideal (f_1, f_2, f_3), namely the first bigraded Betti numbers. The intermediary between singularity types and Rees algebras is once again the syzygy matrix F, or rather a matrix of linear forms derived from it.

This is a report on joint work with D. Cox, A. Kustin, and B. Ulrich, and with A. Kustin and B. Ulrich, respectively.