Searching for structural reasons behind old results and conjectures of
Chudnovksy regarding the least degree of a nonzero form in an ideal of fat
points in projective N-space, we make conjectures which explain them, and we
prove the conjectures in certain cases, including the case of general points in
the projective plane. Our conjectures were also partly motivated by the
Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of
prime ideals in local rings, but in contrast we consider higher symbolic powers
of homogeneous ideals in polynomial rings.Comment: 13 pages; for version 3 a minor change was made to the
acknowledgments but no change was made to mathematical content; for version 2
a reference to a paper of Esnault and Viehweg has been added; related to this
a new section, 4.2, has been included with additional questions. Otherwise,
version 2 is the same as version
