The Sliding Scale Conjecture was posed by Bellare, Goldwasser, Lund and Russell in 1993 and has been open since. It says that there are PCPs with constant number of queries, polynomial alphabet and polynomially small error. We show that the conjecture can be proved assuming a certain geometric conjecture about curves over finite fields.

The geometric conjecture states that there are small families of low degree curves that behave, both in their distribution over points and in the intersections between pairs of curves from the family, similarly to the family of all low degree curves.