Normality, equations and syzygies of secant varieties, I

I will discuss the geometry of the variety of secant lines to a smooth projective variety. Using the work of Bertram and Thaddeus on the moduli of rank 2 vector bundles as a guide, I will show how one can quickly understand the singularities of secant varieties. I will then discuss joint work with Jessica Sidman regarding the regularity, normality, and Cohen-Macaulayness of the secant variety to a smooth curve.