Linear systems with multiple base points in P2

By Brian Harbourne, Joaquim Roé. 2001

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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, for certain $m$ and $n$ (which in certain cases can both be large), the Hilbert function and minimal free resolution are determined for symbolic powers $I^{(m)}$ for the ideal $I$ defining $n$ general points of ${\bf CP}^2$, in agreement with conjectures made by Nagata, Segre, Harbourne, Gimigliano and Hirschowitz. With the exception of Evain's determination of the Hilbert function when $n$ is a power of 4 and Harbourne-Holay-Fitchett's resulting determination of the resolution when, in addition, $m$ is sufficiently large, all previous such results assume either $m$ or $n$ is small (e.g., $m\le 12$ or $n\le 9$).

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

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