Namespaces
Variants
Actions

Difference between revisions of "Arithmetic genus"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (MR/ZBL numbers added)
Line 28: Line 28:
  
 
====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>
+
<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>

Revision as of 21:50, 30 March 2012

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
How to Cite This Entry:
Arithmetic genus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_genus&oldid=23757
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article