Generators for Symbolic Powers of Ideals Defining General Points of $P^2$

Abstract

Given distinct points $p_1,\cdots,p_r$ of the projective plane $P^2$ and a positive integer $m$, the homogeneous ideal defining the fat point subscheme $Z=m(p_1+\cdots+p_r)$ is the symbolic power $I^{(m)}$ of the homogeneous ideal $I$ defining the smooth union of the $r$ points $p_1,\ldots,p_r$. If $p_1,\ldots,p_r$ are sufficiently general, it is known that the maximal rank conjecture holds for $I$; i.e., for every $d$ the multiplication map $I_1\otimes I_d\to I_{(d+1)}$ 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