Nagata's conjecture on curves

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In mathematics, the Nagata conjecture on curves governs the minimal degree required for a plane algebraic curve to pass though a collection of very general points with prescribed multiplicity.

More precisely suppose p_1,\ldots,p_r are very general points in the projective plane P2 and that m_1,\ldots,m_r are given positive integers. The Nagata conjecture states that for r\ge 9 any curve C in P2 that passes through each of the points pi with multiplicity mi must satisfy

\mbox{deg} C > {\sum_{i=1}^r m_i \over \sqrt{r}}

The only case when this is known to hold is when r is a perfect square (i.e. is of the form r = s2 for some integer s which was proved by Nagata. Despite much interest the other cases remain open. A more modern formulation of this conjecture is often given in terms of Seshadri constants and has been generalised to other surfaces under the name of the Nagata-Biran conjecture.

The condition r > 9 is easily seen to be necessary. The cases r\ge 9 and r < 9 are distinguished by whether or not the canonical bundle on the blowup of P2 at a collection of r points is ample.

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