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 k[x1, ..., xn] over some field 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.
Statementedit
Nagata Conjecture. Suppose p1, ..., pr are very general points in P2 and that m1, ..., mr are given positive integers. Then for r > 9 any curve C in P2 that passes through each of the points pi with multiplicity mi must satisfy
The condition r > 9 is necessary: The cases r > 9 and r ≤ 9 are distinguished by whether or not the anti-canonical bundle on the blowup of P2 at a collection of r points is nef. In the case where r ≤ 9, the cone theorem essentially gives a complete description of the cone of curves of the blow-up of the plane.
Current statusedit
The only case when this is known to hold is when 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.
Referencesedit
Harbourne, Brian (2001), "On Nagata's conjecture", Journal of Algebra, 236 (2): 692–702, arXiv:math/9909093, doi:10.1006/jabr.2000.8515, MR 1813496.
Nagata, Masayoshi (1959), "On the 14-th problem of Hilbert", American Journal of Mathematics, 81 (3): 766–772, doi:10.2307/2372927, JSTOR 2372927, MR 0105409.
Strycharz-Szemberg, Beata; Szemberg, Tomasz (2004), "Remarks on the Nagata conjecture", Serdica Mathematical Journal, 30 (2–3): 405–430, hdl:10525/1746, MR 2098342.
April 11, 2024
nagata, conjecture, curves, conjecture, about, wild, automorphisms, nagata, conjecture, mathematics, nagata, conjecture, curves, named, after, masayoshi, nagata, governs, minimal, degree, required, plane, algebraic, curve, pass, through, collection, very, gene. For the conjecture about wild automorphisms see 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 Contents 1 History 2 Statement 3 Current status 4 ReferencesHistory editNagata 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 k x1 xn over some field 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 editNagata Conjecture Suppose p1 pr are very general points in P2 and that m1 mr are given positive integers Then for r gt 9 any curve C in P2 that passes through each of the points pi with multiplicity mi must satisfydeg C gt 1r i 1rmi displaystyle deg C gt frac 1 sqrt r sum i 1 r m i nbsp dd The condition r gt 9 is necessary The cases r gt 9 and r 9 are distinguished by whether or not the anti canonical bundle on the blowup of P2 at a collection of r points is nef In the case where r 9 the cone theorem essentially gives a complete description of the cone of curves of the blow up of the plane Current status editThe only case when this is known to hold is when 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 editHarbourne Brian 2001 On Nagata s conjecture Journal of Algebra 236 2 692 702 arXiv math 9909093 doi 10 1006 jabr 2000 8515 MR 1813496 Nagata Masayoshi 1959 On the 14 th problem of Hilbert American Journal of Mathematics 81 3 766 772 doi 10 2307 2372927 JSTOR 2372927 MR 0105409 Strycharz Szemberg Beata Szemberg Tomasz 2004 Remarks on the Nagata conjecture Serdica Mathematical Journal 30 2 3 405 430 hdl 10525 1746 MR 2098342 Retrieved from https en wikipedia org w index php title Nagata 27s conjecture on curves amp oldid 1023693538, wikipedia, wiki, book, books, library,
