Assumption for computing the genus of projective curves
Let $n\in \mathbb{N}$ and let $I$ be a homogenous prime ideal of $\mathbb{Q}[x_0,...,x_n]$ such that $X=V_+(I)$ is of $\dim=1$. I am coding in the Sage, based on the "[Interface to Singular]", using the function
I.genus()
to compute the genus of $X=V_+(I)$. Such function refers to the "[Singular]".
Here, I search the [genus computation in Singular], and by their description, the genus of $X$ means the arithmetic genus of the normalization of $X$. However, it is not clear to me what is the assumption of the projective curve in this function.
More precisely, my question is, do we have any assumption of $X$? For example, do we require $X$ is geometrically irreducible (i.e., $X$ base change to $\bar{\mathbb{Q}}$ is still irreducible)? Or do we require $X$ is a plane (i.e., $n=2$) or space (i.e., $n=3$) curve?
