Equivalence Relations in Algebraic Geometry

1. The cycle groups C8. An algebraic variety F i n w-dimensional complex projective space P ( n ) is obtained by equating to zero a finite number of forms Fi(x0, • • • , xn)f • • • , Fm(xo, • • • , xn) with complex coefficients ; F is assumed to be nonempty. If F i s irreducible, that is, if V is not the union of a finite number of proper subvarieties, it is possible to associate with V in several ways a complex dimension d. For example, just as P ( 1 ) is topologically equivalent to a real 2dimensional sphere, so can every P ( n ) be represented topologically by a 2w-dimensional real complex in the sense of combinatorial topology. (See [ l ] ; numbers in brackets refer to the references.) In this representation, F goes over into an even-dimensional, connected, orientable, closed complex whose dimension is defined as 2d, This complex is denoted by K^ and F itself by F (d ) . Consider the set T8 of irreducible, s-dimensional subvarieties of F (d ) for some fixed s, where OSs^d. A function on T8 is called integral if its value for every element of T8 is a rational integer, and if the function is zero except for a t most a finite number of elements of T8; these functions constitute of course an additive group, denoted by C8. We identify the integral function which at the elements Wi*\ • • , W$ of T8 assumes the values wi, • • • , tin and which is zero everywhere else on T8 with the linear combination niWi^-jr • • ' +nhWJf\ Since every W® gives rise to a 2s-dimensional, connected, closed, orientable subcomplex of K\ the above linear combination can be interpreted as a 2s-dimensional cycle of K in the sense of topology. This fact is the reason why we call the elements of C8 the s-dimensional cycles of F (d ) and often consider C8 as a subgroup of the 2s-dimensional cycle group of K. A cycle is called effective if, considered as a function, it never assumes a negative value; otherwise the cycle is called virtual. The effective cycles are clearly closed under addition but not under subtraction, and every cycle is the difference of two effective cycles. The group C8 is completely determined by the cardinal number of r „ and hence its structure is of no interest. The importance of C8 lies in the fact that the different aspects of the geometry of F (d) are most conveniently studied by means of the equivalence relations which