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A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440001.png" />. There are two different genera — the [[Arithmetic genus|arithmetic genus]] and the [[Geometric genus|geometric genus]]. The geometric genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440002.png" /> of a complete smooth algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440003.png" /> is equal to
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<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/g/g044/g044000/g0440004.png" /></td> </tr></table>
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i.e. to the dimension of the space of regular differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440005.png" />-forms (cf. [[Differential form|Differential form]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440006.png" />. The arithmetic genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440007.png" /> of a complete smooth algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g0440008.png" /> is equal to
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A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field  $  k $.  
 +
There are two different genera — the [[Arithmetic genus|arithmetic genus]] and the [[Geometric genus|geometric genus]]. The geometric genus $  p _ {g} $
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of a complete smooth algebraic surface $  X $
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is equal to
  
<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/g/g044/g044000/g0440009.png" /></td> </tr></table>
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$$
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p _ {g}  =   \mathop{\rm dim} _ {g}  H  ^ {0} ( X , \Omega _ {X}  ^ {2} ) ,
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$$
  
The geometric and arithmetic genera of a complete smooth algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g04400010.png" /> are related by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g04400011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g04400012.png" /> is the [[Irregularity|irregularity]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g04400013.png" />, which is equal to the dimension of the space of regular differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g04400014.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044000/g04400015.png" />.
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i.e. to the dimension of the space of regular differential  $  2 $-
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forms (cf. [[Differential form|Differential form]]) on  $  X $.
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The arithmetic genus  $  p _ {a} $
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of a complete smooth algebraic surface  $  X $
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is equal to
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$$
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p _ {a}  =  \chi ( X , {\mathcal O} _ {X} ) - 1  = \
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\mathop{\rm dim} _ {k}  H  ^ {2} ( X , {\mathcal O} _ {X} ) -
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\mathop{\rm dim} _ {k}  H  ^ {1} ( X , {\mathcal O} _ {X} ) .
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$$
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The geometric and arithmetic genera of a complete smooth algebraic surface $  X $
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are related by the formula $  p _ {g} - p _ {a} = q $,  
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where $  q $
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is the [[Irregularity|irregularity]] of $  X $,  
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which is equal to the dimension of the space of regular differential $  1 $-
 +
forms on $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R., et al. Shafarevich, "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|1392959}} {{MR|1060325}} {{ZBL|0830.00008}} {{ZBL|0733.14015}} {{ZBL|0832.14026}} {{ZBL|0509.14036}} {{ZBL|0492.14024}} {{ZBL|0379.14006}} {{ZBL|0253.14006}} {{ZBL|0154.21001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R., et al. Shafarevich, "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|1392959}} {{MR|1060325}} {{ZBL|0830.00008}} {{ZBL|0733.14015}} {{ZBL|0832.14026}} {{ZBL|0509.14036}} {{ZBL|0492.14024}} {{ZBL|0379.14006}} {{ZBL|0253.14006}} {{ZBL|0154.21001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>

Revision as of 19:41, 5 June 2020


A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field $ k $. There are two different genera — the arithmetic genus and the geometric genus. The geometric genus $ p _ {g} $ of a complete smooth algebraic surface $ X $ is equal to

$$ p _ {g} = \mathop{\rm dim} _ {g} H ^ {0} ( X , \Omega _ {X} ^ {2} ) , $$

i.e. to the dimension of the space of regular differential $ 2 $- forms (cf. Differential form) on $ X $. The arithmetic genus $ p _ {a} $ of a complete smooth algebraic surface $ X $ is equal to

$$ p _ {a} = \chi ( X , {\mathcal O} _ {X} ) - 1 = \ \mathop{\rm dim} _ {k} H ^ {2} ( X , {\mathcal O} _ {X} ) - \mathop{\rm dim} _ {k} H ^ {1} ( X , {\mathcal O} _ {X} ) . $$

The geometric and arithmetic genera of a complete smooth algebraic surface $ X $ are related by the formula $ p _ {g} - p _ {a} = q $, where $ q $ is the irregularity of $ X $, which is equal to the dimension of the space of regular differential $ 1 $- forms on $ X $.

References

[1] I.R., et al. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001

Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[a2] A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023
[a3] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
How to Cite This Entry:
Genus of a surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_surface&oldid=47081
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article