An Algorithm for Fat Points on P2

Abstract

Let $F$ be a line bundle on the blow-up $X$ of $P^2$ at $r$ general points $p_1, ..., p_r$ and let $L$ be the pullback to $X$ of the line bundle coming from a line on $P^2$. Under reasonable hypotheses that are conjectured always to hold if the points $p_1, ..., p_r$ are sufficiently general, it is shown that the computation of the dimension of the cokernel of the natural map $μ_F:Γ(F)\otimesΓ(L)\toΓ(F\otimes L)$ reduces to the case that $F$ is ample. As an application, a complete determination of the dimension of the cokernel of $μ_F$ is obtained when $r\le 7$, thereby solving the Ideal Generation Problem for fat point subschemes involving up to 7 general points of the plane and giving an algorithm depending only on the multiplicities $m_i$ for determining the modules in a minimal free resolution of the ideal defining a fat point subscheme $m_1p_1+...+ m_7p_7$ for general points $p_i$. All results hold for an arbitrary algebraically closed ground field $k$.