Nagata's conjecture on curves

{{for|the conjecture about wild automorphisms|Nagata's conjecture}}

In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities.

History

Nagata arrived at the conjecture via work on the 14th problem of Hilbert, which asks whether the invariant ring of a linear group action on the polynomial ring {{math|k[x₁, ..., x_n]}} over some field {{mvar|k}} is finitely generated. Nagata published the conjecture in a 1959 paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem.

Statement

:Nagata Conjecture. Suppose {{math|p₁, ..., p_r}} are very general points in {{math|P²}} and that {{math|m₁, ..., m_r}} are given positive integers. Then for {{math|r > 9}} any curve {{mvar|C}} in {{math|P²}} that passes through each of the points {{math|p_i}} with multiplicity {{math|m_i}} must satisfy

:: $\deg C > \frac{1}{\sqrt{r}}\sum_{i=1}^r m_i.$

The condition {{math|r > 9}} is necessary: The cases {{math|r > 9}} and {{math|r ≤ 9}} are distinguished by whether or not the anti-canonical bundle on the blowup of {{math|P²}} at a collection of {{mvar|r}} points is nef. In the case where {{math|r ≤ 9}}, the cone theorem essentially gives a complete description of the cone of curves of the blow-up of the plane.

Current status

The only case when this is known to hold is when {{mvar|r}} is a perfect square, 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.

References

{{citation

| last = Harbourne | first = Brian | author-link=Brian Harbourne (mathematician)

| doi = 10.1006/jabr.2000.8515

| issue = 2

| journal = Journal of Algebra

| mr = 1813496

| pages = 692–702

| title = On Nagata's conjecture

| volume = 236

| year = 2001| arxiv = math/9909093

}}.

{{citation

| last = Nagata | first = Masayoshi

| journal = American Journal of Mathematics

| jstor = 2372927

| mr = 0105409

| pages = 766–772

| title = On the 14-th problem of Hilbert

| volume = 81

| year = 1959

| issue = 3

| doi=10.2307/2372927}}.

{{citation

| last1 = Strycharz-Szemberg | first1 = Beata

| last2 = Szemberg | first2 = Tomasz | author2-link=Tomasz Szemberg (mathematician)

| issue = 2–3

| journal = Serdica Mathematical Journal

| mr = 2098342

| pages = 405–430

| title = Remarks on the Nagata conjecture

| volume = 30

| year = 2004| hdl = 10525/1746

}}.

Category:Algebraic curves

Category:Conjectures