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 $\phi(I,0)$ of the Hilbert polynomial $\phi(I,m)$ of $I$ by the formula

$$p_a(X)=(-1)^n(\phi(I,0)-1).$$

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

$$p_a(X)=(-1)^n(\chi(X,\mathcal O_X)-1),$$

where

$$\chi(X,\mathcal O_X)=\sum_{i=0}^n(-1)^i\dim_kH^i(X,\mathcal O_X)$$

is the Euler characteristic of the variety $X$ with coefficients in the structure sheaf $\mathcal O_X$. 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 $k=\mathbf C$ 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 $\mathcal O_X(D)$ 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 $n\leq3$ only.

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