Fujita's Conjecture and Frobenius amplitude

We prove a version of Fujita's Conjecture in arbitrary characteristic, generalizing results of K.E. Smith. Our methods use the Frobenius morphism, but avoid tight closure theory. We also obtain versions of Fujita's Conjecture for coherent sheaves with certain ampleness properties.Comment: 8 pages. Erratum added to replace Lemma 2.

Criteria for \sigma-ampleness

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $\sigma$-ample divisor, where $\sigma$ is an automorphism of a projective scheme X. Many open questions regarding $\sigma$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $\sigma$-ample. As a consequence, we show right and left $\sigma$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $\sigma$ yield a $\sigma$-ample divisor.Comment: 16 pages, LaTeX2e, to appear in J. of the AMS, minor errors corrected (esp. in 1.4 and 3.1), proofs simplifie

Ample filters and Frobenius amplitude

Let $X$ be a projective scheme over a field. We show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the Frobenius morphism. We also compare various definitions of ample locally free sheaf and show that the vanishing given by the Frobenius morphism is, in a certain sense, the strongest possible. Our work can be viewed as various generalizations of the Serre Vanishing Theorem.Comment: 15 pages, major improvement in results, typo fixed in Equation 2.5, warning footnote added to Lemma 2.

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

Naive Noncommutative Blowing Up

Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X > 1. Assume that c in X and s in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R=R(X,c,L,s) with surprising properties. In particular: (1) R is always noetherian but never strongly noetherian. (2) If R is generated in degree one then the images of the R-point modules in qgr(R) are naturally in (1-1) correspondence with the closed points of X. However, both in qgr(R) and in gr(R), the R-point modules are not parametrized by a projective scheme. (3) qgr R has finite cohomological dimension yet H^1(R) is infinite dimensional. This gives a more geometric approach to results of the second author who proved similar results for X=P^n by algebraic methods.Comment: Latex, 42 page

The Cloze Procedure as a Reinforcement Technique for Content Vocabulary

The purpose of this study was to investigate the cloze procedure as a teaching technique for seventh grade science vocabulary. A quasi-experimental, nonrandomized, control group, pretest-posttest design was used for the study. The sample consisted of 41 students (two classes) taught by the same instructor and was equated in terms of reading levels, ages and IQ scores. One class was randomly assigned to control group status and used a variety of vocabulary exercises such as crossword puzzles, word jumbles, acrostics, categorization exercises and word searches to reinforce the content terms. The other class was the experimental group and used a variety of cloze activities to reinforce the same science vocabulary. The students were pretested on 89 words from the ecology unit in the textbook Interaction of Man and the Biosphere published by the Rand McNally Company. Those words which 85 percent of the students had correct were eliminated from the study. This method left 76 core words to be taught during the treatment period. The 76 core words were organized into eight blocks for ease of instruction. After the instructor presented the material which included all the Block I words, students practiced using the core vocabulary by working on the Block I vocabulary activities designed by the experimenter. The control group used a variety of vocabulary tasks while the experimental group used cloze tasks. This procedure was followed for all eight blocks during the treatment period. Upon completion of the five week treatment period, students were posttested. A t-test and an unweighted means solution of a two-way factorial (nonorthogonal) design were used to analyze the data at a .05 level of significance. The results indicated that overall vocabulary mean gain scores and mean posttest scores were not significantly different between the cloze and vocabulary activities group. However, the cloze group scores were consistently higher in both areas. Vocabulary mean gain scores were not substantially different between males and females but cloze males did perform significantly better than the vocabulary activities males. Females displayed an ability to perform equally well with both instructional methods. Recommendations for classroom use of the cloze procedure as well as suggestions for future research were given

On the stability of fully nonlinear hydraulic-fall solutions to the forced water-wave problem

Two-dimensional free-surface flow over localised topography is examined with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite element method. The Hamiltonian structure of the problem is demonstrated and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that the flow ultimately settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction