1995/09/01 by Brian Harbourne, Harbourne, Brian
Mathematics · #14C99 #Algebraic Geometry (math.AG) #FOS: Mathematics #alg-geom #math.AG #msc:14C99
paper · pdf · doi:10.48550/arxiv.alg-geom/9509003
plain tex, 11 pp. The preprint itself is not contained in the Duke archive; plainTeX textfile and dvi versions of this preprint and related software (as described in the abstract) can instead be obtained via the author's www site, http://www.math.unl.edu/~bharbour/ . Comments and requests can be directed to [email protected]
arxiv created 1995/09/01 · arxiv updated 2011/11/09
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