Calculating of genus of a curve

Question

Let $C$ be a curve over $\mathbb{F}_q$ in projective plane. So $C$ can be done as zeroes of some gomogeneous polynomial $\in \mathbb{F}_q[x,y,z]$ with degree $n$. Whether is there algorithm which is polynomial time in $n$ that calculate arithmetic genus of $C$?

Answer 1

Since you have clarified your question, I can now answer.

The arithmetic genus depends only on the degree of $C$ and the dimension of the projective space it is embedded in.

In fact, the arithmetic genus is defined as $$ 1-P_C(0), $$ where $P_C(t)$ is the Hilbert polynomial of $C$.

But the Hilbert polynomial of a curve og degree $d$ in $\mathbb P^2$ can be computed by the exact sequence $$ 0 \to \mathscr O_{\mathbb P^2} (-d) \to \mathscr O_{\mathbb P^2} \to \mathscr O_C\to 0. $$

By additivity of exact sequences, we have $$ P_C(t) = \binom{t+2}{2} -\binom{t-d+2}{2} = td -\frac 12 d^2+\frac 32 d. $$

Thus, we see that the arithmetic genus of any degree $d$ curve (singular, reducible...) is given by $$ 1+\frac 12 d^2 -\frac 32 d = \frac{(d-2)(d-1)}{2}. $$

In particular, a curve of degree $3$ have arithmetic genus $1$.