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
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This classical definition is due to F. Severi [1]. In the general case it is equivalent to the following definition:
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where
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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
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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
,
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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
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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) |
Arithmetic genus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_genus&oldid=14873