On the compactification of Witt group schemes and the deformation of Art theory

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Already we have showed the existence of group schemes which gave the deformations of the group schemes of Witt vectors to tori. Using those group schemes we could contract the unified Kummer-Artin-Schreier-Witt theory. But, when we want to apply the theory to some problems, for example, the lifting problem of cyclic coverings of algebraic curves, partially solved by Green-Matignon, we need more explict description of the group schemes. In 1999 and 2000, we devoted ourselves to construct concretely the group schemes giving the deformations of the group schemes of Witt vectors to tori, and we succeeded to descrive such group schemes by using several Witt vectors. In the background, there is the Cartier thory, and our thory is given by the representation of that by virtue of deformed Artin-Hasse exponential series.To descrive the ramifications of cyclic coverings, we need to compactfy such group schemes. In positive characteristic case. Garuti gave a nice compactifications of group schemes of Witt vectors by means of ruled surfaces. We tried to give compactifications of the deformed group schemes also, even it is in two-dimensional case, and we are on the way to investigate the description of ramification locuses geometrically.