Equations, Syzygies of Projective Varieties, and Varieties of Secant Lines

I will discuss the relationship between the geometry of an embedded projective variety, the equations defining the variety, and the variety of secant lines. In particular, I will outline the work of Green and Lazarsfeld on syzygies of the equations, Hartshorne's conjecture on complete intersections, and Zak's theorem on linear normality.