Editing Genus (mathematics) (section)

==Algebraic geometry==
There are two related definitions of '''genus''' of any [[projective scheme|projective]] algebraic [[Scheme (mathematics)|scheme]] <math>X</math>: the [[arithmetic genus]] and the [[geometric genus]].<ref>{{cite book | last=Hirzebruch | first=Friedrich | author-link=Friedrich Hirzebruch | title=Topological methods in algebraic geometry | others=Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel | edition=Reprint of the 2nd, corr. print. of the 3rd | orig-year=1978 | series=Classics in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | isbn=978-3-540-58663-0 | zbl=0843.14009 }}</ref>  When <math>X</math> is an [[algebraic curve]] with [[Field (mathematics)|field]] of definition the [[complex number]]s, and if <math>X</math> has no [[tangent space|singular points]], then these definitions agree and coincide with the topological definition applied to the [[Riemann surface]] of <math>X</math> (its [[manifold]] of complex points). For example, the definition of [[elliptic curve]] from [[algebraic geometry]] is ''connected non-singular projective curve of genus 1 with a given [[rational point]] on it''.

By the [[Riemann–Roch theorem#Applications|Riemann–Roch theorem]], an irreducible plane curve of degree <math>d</math> given by the vanishing locus of a section <math>s \in \Gamma(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(d))</math> has geometric genus

:<math>g=\frac{(d-1)(d-2)}{2}-s,</math>

where <math>s</math> is the number of singularities when properly counted.