The Stacks project

Exercise 111.61.1 (Definitions). Provide brief definitions of the italicized concepts. Let $k$ be an algebraically closed field. Let $X$ be a projective curve over $k$.

1. a smooth algebra over $k$,

2. the degree of an invertible $\mathcal{O}_ X$-module on $X$,

3. the genus of $X$,

4. the Weil divisor class group of $X$,

5. $X$ is hyperelliptic, and

6. the intersection number of two curves on a smooth projective surface over $k$.