Linear systems with multiple base points in P2

TL;DR

This paper establishes bounds on the degrees of plane curves passing through multiple general points with specified multiplicities, improving previous results and determining Hilbert functions and resolutions for symbolic powers of ideals of these points.

Contribution

It provides new bounds for degrees of curves with multiple base points and determines Hilbert functions and minimal free resolutions for symbolic powers of ideals of general points in P2.

Findings

Bounds for least degree t with multiplicity conditions

Determination of Hilbert functions for symbolic powers

Explicit minimal free resolutions for certain ideals

Abstract

Given positive integers $m_1, m_2, ..., m_n$, and $n$ general points $p_i$ of ${\bf CP}^2$, bounds are given for the least degree $t$ among plane curves passing through each point $p_i$ with multiplicity at least $m_i$, and for the least $t$ such that the $n$ multiple points impose independent conditions on curves of degree $t$, often improving substantially what was previously known. As an application, the Hilbert function (resp., minimal free resolution) is determined for symbolic powers $I^{(m)}$ for the ideal $I$ defining $n$ general points of ${\bf CP}^2$ for infinitely many m for each square n (resp., for infinitely many m for each even square n). Four graphs are included showing other values of m and n for which results are given.