Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Arithmetic Genus And Intersection Number

Algebraic Geometry
Tsakanikas Nickos
Community Team
Articles: 0
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

### Arithmetic Genus And Intersection Number

Let $X$ be a non-singular projective surface over an algebraically closed field $\mathbb{K}$. If $D$ is an effective divisor on $X$, $p_{a} = 1 - \chi(\mathcal{O}_{D})$ is its arithmetic genus and $K$ is a canonical divisor on $X$, show that $2p_{a} - 2 = D.(D+K)$