View source for Arithmetic genus

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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This classical definition is due to F. Severi [[#References|[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 <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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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 <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" />.

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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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.

====References====
<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>