On the uniformity of zero-dimensional complete intersections

After showing that the General Cayley-Bacharach Conjecture formulated by D. Eisenbud, M. Green, and J. Harris (1996) [6] is equivalent to a conjecture about the region of uniformity of a zero-dimensional complete intersection, we prove this conjecture in a number of special cases. In particular, after splitting the conjecture into several intervals, we prove it for the first, the last and part of the penultimate interval. Moreover, we generalize the uniformity results of J. Hansen (2003) [12] and L. Gold, J. Little, and H. Schenck (2005) [9] to level schemes and apply them to obtain bounds for the minimal distance of generalized Reed-Muller codes.