On the Uniformity of Zero-dimensional Complete Intersections

After showing that the General Cayley\u2013Bacharach Conjecture formulated by D. Eisenbud, M. Green, and J. Harris (1996) [6] is equivalent to a conjecture about the region of uniformity of a zerodimensional 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 uniformityresults of J. Hansen (2003) [12] and L. Gold, J. Little, and H. Schenck (2005) [9] to level schemes and apply them to obtain bounds forthe minimal distance of generalized Reed\u2013Muller codes