Nagatas conjecture on curves

The Nagata conjecture on curves, proposed by Masayoshi Nagata, concerns the minimal degree of a plane algebraic curve passing through a set of very general points with specified multiplicities. Nagata's interest in this problem arose from his work on Hilbert's 14th problem, which asks whether the invariant ring of a linear group action is finitely generated. In 1959, Nagata published a counterexample to Hilbert's 14th problem in the American Journal of Mathematics, where he introduced the conjecture.

The conjecture states that for r > 9 points in P² with given multiplicities m₁,...,mᵣ, any curve passing through these points must have a degree exceeding (1/√r) times the sum of the multiplicities. The condition r > 9 is crucial because it relates to whether the anti-canonical bundle on the blowup of P² at r points is nef. For r ≤ 9, the cone theorem provides a complete description of the cone of curves in this context.

Currently, the conjecture is proven only when r is a perfect square, as demonstrated by Nagata. Other cases remain unresolved. Modern formulations often connect the conjecture to Seshadri constants and have been generalized to other surfaces under the name of the Nagata–Biran conjecture.

References include works by Harbourne (2001), Nagata (1959), and Strycharz-Szemberg & Szemberg (2004).