Resolutions of ideals of quasiuniform fat point subschemes of $\mathbf P^2$

The notion of a quasiuniform fat point subscheme $Z\subset\mathbf P^2$is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal $I$ defining $Z$ are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the $m$th symbolic power $I(m;n)$ of an ideal defining $n$ general points of $\mathbf P^2$ when both $m$ and $n$ are large (in particular, for infinitely many $m$ for each of infinitely many $n$, and for infinitely many $n$ for every $m>2$). Resolutions in other cases, such as ``fat points with tails'', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field $k$. As an incidental result, a bound for the regularity of $I(m;n)$ is given which is often a significant improvement on previously known bounds.