Severi Varieties and Sharp Bounds of Castelnuovo-Mumford Regularities

In the first part of the thesis, we give a new proof of Zak's classification theorem of Severi varieties based on Zak's analysis of smooth quadrics on Severi varieties and classification of aCM bundles on smooth quadrics. In the second part of the thesis, we show that a bound of the Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety X in a projective space of dimension r is sharp exactly for complete intersections, provided the variety X is cut out scheme-theoretically by several hypersurfaces in the projective space of dimension r.