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| | Royal Votaw is my name but I never really favored that name. Delaware is the only place I've been residing in. Bookkeeping is what she does. What she loves doing is taking part in croquet and she is attempting to make it a occupation.<br><br>Have a look at my web blog [http://Www.Holzkunst-Scholz.de/index.php?mod=users&action=view&id=7124 extended car warranty] |
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| In [[mathematics]], the '''Nagata conjecture''' on curves, named after [[Masayoshi Nagata]], governs the minimal degree required for a [[Algebraic curve|plane algebraic curve]] to pass through a collection of very general points with prescribed [[Multiplicity (mathematics)|multiplicities]]. Nagata arrived at the conjecture via work on the [[Hilbert's problems|14th problem of Hilbert]], which asks whether the invariant ring of a linear group action on the polynomial ring <math>k[x_1, \ldots x_n]</math> over some field <math>k</math> is [[Finitely generated group|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.
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| More precisely suppose <math>p_1,\ldots,p_r</math> are very general points in the [[projective plane]] <math>P^2</math> and that <math>m_1,\ldots,m_r</math> are given positive integers. The Nagata conjecture states that for <math>r > 9</math> any curve <math>C</math> in <math>P^2</math> that passes through each of the points <math>p_i</math> with multiplicity <math>m_i</math> must satisfy
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| :<math>\mathrm{deg}\, C > {\sum_{i=1}^r m_i \over \sqrt{r}}.</math>
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| The only case when this is known to hold is when <math>r</math> is a perfect square (i.e. is of the form <math>r=s^2</math> for some integer <math>s</math>), which was proved by [[Masayoshi Nagata|Nagata]]. Despite much interest the other cases remain open. A more modern formulation of this conjecture is often given in terms of [[Seshadri constant]]s and has been generalised to other surfaces under the name of the [[Nagata–Biran conjecture]].
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| The condition <math>r> 9</math> is easily seen to be necessary. The cases <math>r> 9</math> and <math>r \le 9</math> are distinguished by whether or not the [[canonical bundle|anti-canonical bundle]] on the [[Blowing up|blowup]] of <math>P^2</math> at a collection of <math>r</math> points is [[Numerically effective|nef]].
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| {{DEFAULTSORT:Nagata's Conjecture On Curves}}
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| [[Category:Algebraic curves]]
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| [[Category:Conjectures]]
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Latest revision as of 22:58, 19 December 2014
Royal Votaw is my name but I never really favored that name. Delaware is the only place I've been residing in. Bookkeeping is what she does. What she loves doing is taking part in croquet and she is attempting to make it a occupation.
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