THE KANSAS UNIVERSITY 

 SCIENCE BULLETIN. 



Vol. X, No. 1.] January, 1917. ['^l^xx^nTI 



Note on a Problem in the Theory of Algebraic Manifolds. 



BY S. LEFSCHETZ. 



1. In a recent paper' I have solved the following problem: 

 Let Di, D-2, . . . . D^, be algebraic manifolds belonging to an h'- 

 reducible algebraic r-dimensional variety, of which the hyper- 

 surfaces Ci , Co , . . . . C," , form a base. If 



to obtain the genus of D\, Di, . . . . D^, complete intersection of 

 the D's, in terms of the genera of the C's and of their mutual in- 

 tersections. It was shown there that a formula answering the 

 purpose exists, independent from the signs of the integers /, and 

 this formula was explicitly given (formula 15). This was fin- 

 ally applied to loci of spaces, or generalized scrolls, and to com- 

 plete intersections in ordinary r-space. 



The solution obtained was based upon Severi's formula for the 

 genus of a manifold sum of two others'-'. However, the funda- 

 mental formula was derived only by an indirect method. It is 

 the main object of the present note to show how the problem in 

 question can be solved by a complete inversion of Severi's formula, 

 inversion which in itself is not without interest. We will make 

 use throughout of the notion of virtual manifolds considered at 

 length in the note already referred to. With this notion the 

 problem is purely one of algebra, and consists in the extension of 

 a certain algebraic functional operation. The abstract problem 

 underlying the symbolism used is considered in §§ 7, 8. The in- 

 teresting result is obtained that the most general addition for- 



1. The arithmetic genus of an algebraic manifold immersed in another. Annals of Mathe- 

 ;^-^ maties, 1916. Vol. 17, p. 197. 



*"" 2. Fundamenti per la geometria sulle varieta algehriche. Rendiconti del Circolo Mathe- 



O^ matico di Palermo, 1909, p. 42. 



"^ (3) 



