On the Nagata Problem

The mixed capacitated general routing problem with turn penalties

0957-4174/$ see front matter 2011 Elsevier Ltd. A doi:10.1016/j.eswa.2011.04.092 ⇑ Corresponding author. Tel.: +34963877007x7666 E-mail addresses: [email protected] (O. Bräy Martínez), [email protected] (Y. Nagata), dsole In this paper we deal with the mixed capacitated general routing problem with turn penalties. This problem generalizes many important arc and node routing proble...

Segmentation of continuous problem space based on local prediction

On the postulation of s^d fat points in P^d

In connection with his counterexample to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in P 2 and proved it when r is a square. Iarrobino formulated a similar conjecture in P d. We prove Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new counterexamples modeled on those by Nagata.

Counterexamples to the Fourteenth Problem of Hilbert

Let K be a field, K[X] = K[X1, . . . , Xn] the polynomial ring in n variables over K for some n ∈ N, and K(X) the field of fractions of K[X]. Assume that L is a subfield of K(X) containing K. Then, the Fourteenth Problem of Hilbert asks whether the Ksubalgebra L ∩K[X] of K[X] is finitely generated. Zariski [17] showed in 1954 that the answer to this problem is affirmative if the transcendence d...

Seshadri Constants on Symmetric Products of Curves

Let Xg = C (2) g be the second symmetric product of a very general curve of genus g. We reduce the problem of describing the ample cone on Xg to a problem involving the Seshadri constant of a point on Xg−1. Using this we recover a result of Ciliberto-Kouvidakis that reduces finding the ample cone of Xg to the Nagata conjecture when g ≥ 9. We also give new bounds on the the ample cone of Xg when...