Building on work of the fourth author and Morelli’s work, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K K of characteristic zero is a composite of blowings up and blowings down with nonsingular centers.

Regretfully, we work over an algebraically closed field k of characteristic 0. 0.1. The problem. Roughly speaking, the semistable reduction problem we address here asks for the following:Let X → B be a surjective morphism of complex projective varieties with geometrically integral generic fiber. Find a generically finite proper surjective morphism (that is, an alteration) B 1 → B, and a proper birational morphism (that is, a modification)Of course, one needs to decide what a "nice morphism" means.The question was posed, among other places, in the introduction of [KKMS], p. vii. It can be viewed as a natural extension of Hironaka's theorem on resolution of singularities, which is in a sense "the general fiber" of semistable reduction. 0.2. Brief history. The case dim B = 1, dim X = 2 is very old, see [A-W]. When dim B = 1, semistable reduction was obtained in [KKMS], in the best possible sense: Y is nonsingular, and all the fibers are reduced, strict divisors of normal crossings.Using a result of Kawamata on ramified covers (see [Kawa], theorem 17), one can obtain semistable reduction "in codimension 1" over a base of arbitrary dimension. Below, we will refer to the result of Kawamata as "Kawamata's trick". We will discuss it in detail in section 5.The case where dim X = dim B + 1 has recently been proven by de Jong [dJ]. Here one shows that any family of curves can be made into a family of nodal curves, which are indeed as "nice" as one may expect.Using recent difficult results of Alexeev, Kollár and Shepherd-Barron (see [Al], [Al1]), one obtains a version of the case dim X = dim B +2. Here each fiber is a semi-log-canonical surface.Up until recently, not much has been known about the case dim X > dim B + 2. Often one finds remarks of the following flavor: "since we do not have a semistable reduction result over a base of higher dimension, we will work around it in the following technical manner...". 0.3. Definition of semistable families. We give here a description of the best possible kind of morphisms we have in mind.Let f : X → B be a flat morphism of nonsingular projective varieties with connected fibers. Somewhat informally, we say that f is semistable if for each point x ∈ X with f (x) = b there is a choice of formal coordinatesB b = Spec k[[t i ]] andX x = Spec k[[x j ]], such that f is given by:

Abstract-With the current rapid growth in multimedia technology, there is an imminent need for efficient techniclues to search and query large image databases. Because of their unique and peculiar needs, image databases cannot be treated iri a similar fashion to other types of digital libraries. The contextual dependencies present in images, and the complex nature of two-dimensional image data make the representation issues more difficult for image databases. An invariant representation of an image is still an open research issue. For these reasons, it is difficult to find a universal content-based retrieval technique. Current approaches based on shape, texture, and color for indexing image databases have met with limited success. Further, these techniques have not been adequately tested in the presence of noise and distortions. A given application domain offers stronger constraints for improving the retrieval performance. Fingerprint databases are characterized by their large size as well as noisy and distorted query images.Distortions are very common in fingerprint images due to elasticity of the skin. In this paper, a method of indexincl large fingerprint image databases is presented. The approach integrates a number of domain-specific high-level features such as pattern class and ridge density at higher levels of the search. At the lowest level, it incorporates elastic structural feature-based matching for indexing the database. With a multilevel indexing approach, we have been able to reduce the search space. The search engine has also been implemented on Splash 2-a field programmable gate array (FPGA)-based array processor to obtain near-,4SIC level speed of matching. Our approach has been tested on a locally collected test data and on NIST-9, a large fingerprint database available in the public domain. index Terms-Image database, fingerprint matching, minutiae points, image registration, indexing, field programmable gate array.

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