PRESENT PROBLEMS OF GEOMETRY 569 



A question which awaits solution even in the case of the plane 

 is that relating to the invariants of the group of Cremona trans- 

 formations proper. The genus and the moduli of a curve are unaltered 

 by all birational transformations, but the problem arises : Are there 

 properties of curves which remain unchanged by Cremona, although 

 not by other birational transformations ? From the fact that 

 birationally equivalent curves need not be equivalent under the 

 Cremona group, it would seem that such invariants -- Cremona 

 invariants proper --do exist, but no actual examples have yet been 

 obtained. The problem may be restated in the form: What are the 

 necessary and sufficient conditions which must be fulfilled by two 

 curves if they are to be equivalent with respect to Cremona trans- 

 formations? Equality of genera and moduli, as already remarked, is 

 necessary but not sufficient. 



The invariant theory of birational transformations has for its 

 principal object the study of the linear systems of point groups 

 on a given algebraic curve, that is, the point groups cut out by 

 linear systems of curves. Its foundations were implicitly laid by 

 Riemann in his discussion of the equivalent theory of algebraic func- 

 tions on a Riemann surface, though the actual application to curves 

 is due to Clebsch. Most of the later work has proceeded along 

 the algebraic-geometric lines developed by Brill and Noether, the 

 promising purely geometric treatment inaugurated by Segre being 

 rather neglected. 



The extension of this type of geometry to space, that is, the de- 

 velopment of a systematic geometry on a fundamental algebraic 

 surface (especially as regards the linear systems of curves situated 

 thereon), is one of the main tasks of recent mathematics. The 

 geometric treatment is given in the memoirs of Enriques and Castel- 

 nuovo, while the corresponding functional aspect is the subject of 

 the treatise of Picard and Simart on algebraic functions of two 

 variables, at present in course of publication. 



The most interesting feature of the investigations belonging in 

 this field is the often unexpected light which they throw on the 

 inter-relations of distinct fields of mathematics, and the advantage 

 derived from such relations. For example, Picard (as he himself 

 relates on presenting the second volume of his treatise to the Paris 

 Academy a few months ago) * for a long time was unable to prove 

 directly that the integrals of algebraic total differentials can be 

 reduced, in general, to algebraic-logarithmic combinations, until 

 finally a method for deciding the matter was suggested by a theorem 

 on surfaces which Noether had stated some twenty years earlier. 

 Again, in the enumeration of the double integrals of the second 

 species, Picard arrived at a certain result, which was soon noticed 

 1 Comptes Rendus, February 1, 1904. 



