Arithmetic genus

A numerical invariant of algebraic varieties (cf. Algebraic variety). For an arbitrary projective variety $X$ (over a field $k$ ) all irreducible components of which have dimension $n$ , and which is defined by a homogeneous ideal $I$ in the ring $k [T_{0}, \dots, T_{N}]$ , the arithmetic genus $p_{a} (X)$ is expressed using the constant term $ϕ (I, 0)$ of the Hilbert polynomial $ϕ (I, m)$ of $I$ by the formula

$p_{a} (X) = (- 1)^{n} (ϕ (I, 0) - 1) .$

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

$p_{a} (X) = (- 1)^{n} (χ (X, O_{X}) - 1),$

where

$χ (X, O_{X}) = \sum_{i = 0}^{n} (- 1)^{i} \dim_{k} H^{i} (X, O_{X})$

is the Euler characteristic of the variety $X$ with coefficients in the structure sheaf $O_{X}$ . 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 $p_{a} (X)$ relative to biregular mappings. If $X$ is a non-singular connected variety, and $k = C$ is the field of complex numbers, then

$p_{a} (X) = \sum_{i = 0}^{n - 1} g_{n - 1} (X),$

where $g_{k} (X)$ is the dimension of the space of regular differential $k$ -forms on $X$ . Such a definition for $n = 1, 2$ was given by the school of Italian geometers. For example, if $n = 1$ , then $p_{a} (X)$ is the genus of the curve $X$ ; if $n = 2$ ,

$p_{a} (X) = - q + p_{g},$

where $q$ is the irregularity of the surface $X$ , while $p_{g}$ is the geometric genus of $X$ .

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

$p_{a} (D) = p_{a} (D^{'}) .$

The arithmetic genus is a birational invariant in the case of a field $k$ of characteristic zero; in the general case this has so far (1977) been proved for dimensions $n \leq 3$ only.

References[edit]

[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