Arithmetic genus

A numerical invariant of algebraic varieties (cf. Algebraic variety). For an arbitrary projective variety (over a field ) all irreducible components of which have dimension , and which is defined by a homogeneous ideal in the ring , the arithmetic genus is expressed using the constant term of the Hilbert polynomial of by the formula

This classical definition is due to F. Severi [1]. In the general case it is equivalent to the following definition:

where

is the Euler characteristic of the variety with coefficients in the structure sheaf . In this form the definition of the arithmetic genus can be applied to any complete algebraic variety, and this definition also shows the invariance of relative to biregular mappings. If is a non-singular connected variety, and is the field of complex numbers, then

where is the dimension of the space of regular differential -forms on . Such a definition for was given by the school of Italian geometers. For example, if , then is the genus of the curve ; if ,

where is the irregularity of the surface , while is the geometric genus of .

For any divisor on a normal variety , O. Zariski (see [1]) defined the virtual arithmetic genus as the constant term of the Hilbert polynomial of the coherent sheaf corresponding to . If the divisors and are algebraically equivalent, one has

The arithmetic genus is a birational invariant in the case of a field of characteristic zero; in the general case this has so far (1977) been proved for dimensions only.

References

[1]	M. Baldassarri, "Algebraic varieties" , Springer (1956)
[2]	F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German)