Shapes of free resolutions over a local ring

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences.Comment: 14 pages, 1 figure; v2: sections have been reorganized substantially and exposition has been streamline

Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added reference

Non-commutative desingularization of determinantal varieties, I

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.Comment: 52 pages, 3 figures, all comments welcom

Detection of diffuse and specular interface reflections and inter-reflections by color image segmentation

We present a computational model and algorithm for detecting diffuse and specular interface reflections and some inter-reflections. Our color reflection model is based on the dichromatic model for dielectric materials and on a color space, called S space, formed with three orthogonal basis functions. We transform color pixels measured in RGB into the S space and analyze color variations on objects in terms of brightness, hue and saturation which are defined in the S space. When transforming the original RGB data into the S space, we discount the scene illumination color that is estimated using a white reference plate as an active probe. As a result, the color image appears as if the scene illumination is white. Under the whitened illumination, the interface reflection clusters in the S space are all aligned with the brightness direction. The brightness, hue and saturation values exhibit a more direct correspondence to body colors and to diffuse and specular interface reflections, shading, shadows and inter-reflections than the RGB coordinates. We exploit these relationships to segment the color image, and to separate specular and diffuse interface reflections and some inter-reflections from body reflections. The proposed algorithm is effications for uniformly colored dielectric surfaces under singly colored scene illumination. Experimental results conform to our model and algorithm within the liminations discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41303/1/11263_2004_Article_BF00128233.pd