| Abstract: | Fix integers  and let  be the set of all integral, projective and nondegenerate varieties  of degree  and dimension  in the projective space  , such that, for all  ,  does not lie on any variety of dimension  and degree  . We say that a variety  satisfies a flag condition of type  if  belongs to  . In this paper, under the hypotheses  , we determine an upper bound  , depending only on  , for the number  , where  denotes the geometric genus of  . In case  and  , the study of an upper bound for the geometric genus has a quite long history and, for  ,  and  , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data  . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension  in  as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in  . Next we discuss how far  is from  and show a sort of lifting theorem which states that, at least in certain cases, the varieties  of maximal geometric genus  must in fact lie on a flag such as  , where  denotes a subvariety of  of degree  and dimension  . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions. |
