Nagata's conjecture on curves

From Wikipedia, the free encyclopedia

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. 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 $k[x_1, \ldots x_n]$ 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.

More precisely suppose $p_1,\ldots,p_r$ are very general points in the projective plane $P 2$ and that $m_1,\ldots,m_r$ are given positive integers. The Nagata conjecture states that for $r > 9$ any curve $C$ in $P 2$ that passes through each of the points $p i$ with multiplicity $m i$ 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 $r = s 2$ 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 $r > 9$ and $r \le 9$ are distinguished by whether or not the anti-canonical bundle on the blowup of $P 2$ at a collection of $r$ points is nef.

Categories: Algebraic curves | Conjectures

See also ebooksgratis.com: no banners, no cookies, totally FREE.

Nagata's conjecture on curves

From Wikipedia, the free encyclopedia

Views

Navigation

Interaction

Search