Generators for Symbolic Powers of Ideals Defining General Points of P²

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Given distinct points p₁,,pᵣ of the projective plane P² and a positive integer m, the homogeneous ideal defining the fat point subscheme Z=m(p₁++pᵣ) is the symbolic power I⁽ᵐ⁾ of the homogeneous ideal I defining the smooth union of the r points p₁,,pᵣ. If p₁,,pᵣ are sufficiently general, it is known that the maximal rank conjecture holds for I; i.e., for every d the multiplication map I₁⊗ Id→ I₍d₊₁₎ on homogeneous components has maximal rank (meaning the map is either injective or surjective). One easily sees this fails for symbolic powers of ideals defining general points; this preprint relates the failure to the occurrence of (in Nagata's terminology) uniform abnormal curves, and, for r<10, takes complete account of the failure, thereby completely determining the modules in a minimal free resolution of I⁽ᵐ⁾ when r<10. It is also conjectured that maximal rank holds if r>9. Assuming this and a previous conjecture of the author, one can completely determine the modules in a minimal free resolution of I⁽ᵐ⁾ for any r>0 general points and any m>0. The author's www site, http://www.math.unl.edu/ bharbour, makes available, in addition to plainTeX textfile and dvi versions of this preprint, a Macintosh (stuffed and bin hexed) executable and a C source textfile program which output the (conjectural for r>9) modules in a minimal free resolution of I⁽ᵐ⁾ for any r>0 general plane points and any m>0. Web visitors can also run a version of