Genus of a surface
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 Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL |
| [a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Template:ZBL |
| [a2] | A. van de Ven, "Compact complex surfaces" , Springer (1984) MR Template:ZBL |
| [a3] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Template:ZBL |
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