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A numerical invariant of algebraic varieties (cf. [[Algebraic variety|Algebraic variety]]). For an arbitrary projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133101.png" /> (over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133102.png" />) all irreducible components of which have dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133103.png" />, and which is defined by a homogeneous ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133104.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133105.png" />, the arithmetic genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133106.png" /> is expressed using the constant term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133107.png" /> of the [[Hilbert polynomial|Hilbert polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a0133109.png" /> by the formula
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A numerical invariant of algebraic varieties (cf. [[Algebraic variety|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|Hilbert polynomial]] $\phi(I,m)$ of $I$ by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331010.png" /></td> </tr></table>
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$$p_a(X)=(-1)^n(\phi(I,0)-1).$$
  
 
This classical definition is due to F. Severi [[#References|[1]]]. In the general case it is equivalent to the following definition:
 
This classical definition is due to F. Severi [[#References|[1]]]. In the general case it is equivalent to the following definition:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331011.png" /></td> </tr></table>
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$$p_a(X)=(-1)^n(\chi(X,\mathcal O_X)-1),$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331012.png" /></td> </tr></table>
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$$\chi(X,\mathcal O_X)=\sum_{i=0}^n(-1)^i\dim_kH^i(X,\mathcal O_X)$$
  
is the Euler characteristic of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331013.png" /> with coefficients in the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331015.png" /> relative to biregular mappings. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331016.png" /> is a non-singular connected variety, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331017.png" /> is the field of complex numbers, then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331018.png" /></td> </tr></table>
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$$p_a(X)=\sum_{i=0}^{n-1}g_{n-1}(X),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331019.png" /> is the dimension of the space of regular differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331020.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331021.png" />. Such a definition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331022.png" /> was given by the school of Italian geometers. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331024.png" /> is the genus of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331025.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331026.png" />,
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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 $n=2$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331027.png" /></td> </tr></table>
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$$p_a(X)=-q+p_g,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331028.png" /> is the irregularity of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331029.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331030.png" /> is the [[Geometric genus|geometric genus]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331031.png" />.
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where $q$ is the irregularity of the surface $X$, while $p_g$ is the [[Geometric genus|geometric genus]] of $X$.
  
For any divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331032.png" /> on a normal variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331033.png" />, O. Zariski (see [[#References|[1]]]) defined the virtual arithmetic genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331034.png" /> as the constant term of the Hilbert polynomial of the coherent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331035.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331036.png" />. If the divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331038.png" /> are algebraically equivalent, one has
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For any divisor $D$ on a normal variety $X$, O. Zariski (see [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331039.png" /></td> </tr></table>
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$$p_a(D)=p_a(D').$$
  
The arithmetic genus is a birational invariant in the case of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331040.png" /> of characteristic zero; in the general case this has so far (1977) been proved for dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013310/a01331041.png" /> only.
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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.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Baldassarri,   "Algebraic varieties" , Springer (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Hirzebruch,   "Topological methods in algebraic geometry" , Springer (1978) (Translated from German)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR></table>

Latest revision as of 12:41, 19 August 2014

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.

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
How to Cite This Entry:
Arithmetic genus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_genus&oldid=14873
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article