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) MR0082172 Zbl 0995.14003 Zbl 0075.15902 |
[2] | F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) MR1335917 MR0202713 Zbl 0376.14001 |
Arithmetic genus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_genus&oldid=23757