The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds.Comment: 15 pages. v2: Completely rewritten paper. Structure theorem for the cone of nef curves proved in arbitrary dimension using results of Birkar, Cascini, Hacon and McKernan. To appear in Mathematische Zeitschrif

Analysis of the effectiveness of industrial R and D

The criteria used by private industry in evaluating and selecting proposed research and development projects for implementation, and also in determining which R and D facilities are to be acquired were investigated. Conceptual and practical issues inherent in any quantitative analysis of the contribution of R and D to economic growth were identified in order to assist NASA in developing approaches for analzying the economic implication of its own R and D efforts

Inclusion-exclusion and Segre classes

We propose a variation of the notion of Segre class, by forcing a naive `inclusion-exclusion' principle to hold. The resulting class is computationally tractable, and is closely related to Chern-Schwartz-MacPherson classes. We deduce several general properties of the new class from this relation, and obtain an expression for the Milnor class of a scheme in terms of this class.Comment: 8 page

Noncommutative ampleness for multiple divisors

The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Thus we find under natural hypotheses that multi-homogeneous coordinate rings are noetherian and have integer GK-dimension.Comment: 11 pages, LaTeX, minor corrections, to appear in J. Algebr