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We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded., Comment: Final version

We prove a conjecture of Shokurov which characterises toric varieties using log pairs., 40 pages

We prove a conjecture of Starr [11] that the conclusion of bend and break holds for proper algebraic spaces over an algebraically closed field of characteristic zero. The proof uses the minimal model program, rather than reduction modulo $p$ and it even applies to spaces with kawamata log terminal singularities.

This paper seeks to examine the influence of Fabian Socialist thinking as the primary force in the development of critical theory as applied to higher education in Britain. The paper covers the impact of scientific Fabian Socialism and the establishment of the London School of Economics and Political Science, the Frankfurt School and the rise of critical theory and pedagogy, and offers a critique of these perspectives. The social reconstructionist theory, worked out in the USA, posits schools and teachers as planned agents of social and cultural reform by addressing and solving practical social problems. The reconstructionists and critical theorists embrace notions of equality, the eradication of social injustices, multiculturalism, increasing levels of social consciousness and the discussion of controversial issues through employment of critical forms of pedagogy. In Britain, Fabian Socialism led directly to the establishment of the Labour Party as a political entity.

Simons foundation [DMS-1300750, DMS-1265285]; NSF [0701101, 1200656, 1265263]; Simons foundation; Mathematische Forschungsinstitut Oberwolfach; National Science Fund for Distinguished Young Scholars grant from China [11425101]

On Fano Foliations 2.- Rational Curves on Foliated Varieties.- Local Structure of Closed Symmetric 2-Differentials.- Aspects of the Geometry of Varieties with Canonical Singularities.- Geometric Structures and Substructures on Uniruled Projective Manifolds.- Foliations, Shimura Varieties and the Green-Griffiths-Lang Conjecture.- On the Structure of Codimension I Foliations with Pseudoeffective Conormal Bundle.

Any two birational Mori fibre spaces are connected by a sequence of Sarkisov links.

This article is based on the 7th Takagi Lectures that the author delivered at the University of Tokyo on November 21-23, 2009., We explore the circle of ideas connecting finite generation of the Cox ring, Mori dream spaces and invariant theory.

We show that log canonical thresholds satisfy the ACC

Let X be a smooth subvariety of Pr , over an algebraically closed eld k. In an attempt to understand the geometry of X , one would like to understand the singular hyperplane sections of X , and how the family of all hyperplane sections behave around a singular one. Classically this involved the study of in ection points on curves. If X is arbitrary nothing much can be said, so suppose that X is general, meaning that it belongs to an open subset of the Hilbert scheme of Pr . It is natural to expect the singularities of hyperplane sections are isolated and versally deformed by the nearby hyperplane sections. If this property holds, we shall call X hyperplane versal. Versality is de ned in (1.2), and roughly means that every possible deformation appears. Apart from the fact that versality is an interesting property in its own right, there is often a normal form for a versal deformation, and it is a useful property for enumerative questions. We are able to prove the following results

We survey results related to pluricanonical maps of complex projective varieties of general type.

Flips and flops are elementary birational maps which first appear in dimension three. We give examples of how flips and flops appear in many different contexts. We describe the minimal model program and some recent progress centred around the question of termination of flips., National Science Foundation (U.S.) (Grant 0701101), Clay Mathematics Institute

Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that the restriction of L to V is not big. For a smooth variety in characteristic zero, this was proved by Nakamaye using vanishing theorems., Comment: 9 pages; v.2: minor corrections, to appear in Proceedings of the Edinburgh Mathematical Society, volume dedicated to V.V. Shokurov

The ‘objectives model’ of curriculum planning, predicated upon behavioural performances, has become the dominant form of curriculum planning in Europe and elsewhere in the world. This paper argues that the objectives model is satisfactory for training or instruction, but falls down when applied to a true sense of ‘education’. The paper outlines 13 limitations on the use of educational objectives. It is argued that those interested in using objectives are guided by evaluation as assessment rather than principles of procedure for education. Education is about the process of ‘travelling’ on an educational journey – not about ‘arriving’ at a destination. Keywords: instructional objectives, limitations of objectives, curriculum planning, process model. (Published: 1 March 2010) Citation: Education Inquiry Vol. 1, No. 1, March 2010, pp.57–67

Original manuscript September 6, 2012, We show that the number of birational automorphisms of a variety of general type X is bounded by c ⋅ vol(X, K[subscript X]), where c is a constant that only depends on the dimension of X., National Science Foundation (U.S.) (Research Grant 0701101)

We prove the rational connectedness conjecture of V. V. Shokurov in [20] which, in particular, implies that the fibres of a resolution of a variety with divisorial log terminal singularities are rationally chain connected

We prove that there is a fixed constant r=r_n, such that if X is a variety of general type, then the rth pluricanonical map is birational.

The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in Kaehler-Einstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in an appropriately defined sense, the number of dibaryons. Using AdS/CFT backgrounds built from the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999), we show that the gauge theory prediction for the dimension of dibaryonic operators does indeed match the degree of the corresponding holomorphic curves. For AdS/CFT backgrounds built from cones over del Pezzo surfaces, we are able to match the degree of the curves to the conformal dimension of dibaryons for the n'th del Pezzo surface, n=1,2,...,6. Also, for the del Pezzos and the A_k type generalized conifolds, for the dibaryons of smallest conformal dimension, we are able to match the number of holomorphic curves with the number of possible dibaryon operators from gauge theory., Comment: 30 pages, 6 figures, corrected refs; v3 typos corrected

The main result is that a quasi-projective surface has negative log Kodaira dimension (i.e. no log pluricanonical sections) iff it is dominated by images of the affine line. This follows from our main intermediate result, that the smooth locus of a log Fano surface is rationally connected. We also give a classification of all but a bounded family of rank one log del Pezzo surfaces., 159 pages

Since the late 1940s a 'technical rationality' has been advocated for curriculum planners (Tyler, 1949). Curriculum improvement, it was advised, could only count as a 'rational' activity if the planned actions were guided by quite clear behavioural statements of intended educational outcomes that were operationally defined in terms of empirically verifiable changes in pupil behaviour. This model of curriculum designa known as the 'objectives model', has been widely taken up, although curiously curriculum designers in Ireland have resisted its lore. One of the most obvious shortcomings noted by curriculum scholars of Irish education has been the inability of curriculum designers and policy-makers to specify clear and valued intentions for curriculum (Mulcahy, 1981; Crooks & McKernan, 1984; O'Buachalla, 1988; Mulcahy & O'Sullivan, 1989), and to construct a common 'core' curriculum based on an agreed set of outcomes for all pupils. Indeed, curriculum development and innovation has not generally been regarded as a rational activity with either clear objectives or the procedural means for realising such goals or intentions. In recent years, however, there has been a recognition of this impoverished state of affairs by central authorities (CE NCCA, 1989) and a declared commitment to what is generally considered to be rational curriculum planning and assessment. It is not the purpose of this paper to exhume the many criticisms of such work (Elliott, 1988; Simons, 1988). In extreme cases, these can lead to distorted forms of normreferenced testing and even a 'league division table' for schools through national assessment and monitoring of student achievement along the lines developed in the United States and Britain through the various National Assessment of Educational Performance projects, notably the work of the Assessment of Performance Units in the UK (Welford et al., 1986) and the kind of debate brought about by the establishment of 'National Curriculums' in the UK and many other European nations, including Ireland. Rather, it is my intention to describe a conceptual tool which has been constructed to act as a model for curriculum design and evaluation in light of the declared statements made by central educational decision-makers. Thus, the chief outcomes of this paper will be, first, an evaluation grid by which school staff and others can reflect upon the intentions and declared goals of their work; and second, a reaffirmation and commitment to the concept of a core curriculum.

The purpose of the study was to discover the extent to which Catholic (N=125) and Protestant (N=77) post‐primary school teachers participating in a social studies curriculum development project in Northern Ireland were constrained from teaching controversial issues of a social, religious or political nature. A method for ranking teacher, school and community constraints is presented, and the results of a survey of 202 teachers are reported. The results suggest that there is widespread support for handling controversial issues in the curriculum, although Catholic teachers are more favourable to such teaching than Protestant teachers. A high degree of concordance (rho=0.94) was found in ranking constraints when communal groups of teachers are compared. The most important constraints are ‘teacher factors’ such as ‘lack of knowledge or skills’, and ‘teacher's own prejudices’. Teachers regarded communal constraints such as ‘fear of paramilitary groups’ and ‘disapproval of the church’ as having least i...

(1982). Value systems and race relations in Northern Ireland and America. Ethnic and Racial Studies: Vol. 5, No. 2, pp. 156-174.