Suppose $X\subset \mathbb{P}_k^N$ a nonsingular curve is a complete intersection of hypersurfaces $F_1, \cdots, F_{N-1}$ (of degrees $d_1, \cdots, d_{N-1}$ resp). Then, we know that the canonical divisor on $X$ is $\mathcal{O}_X(d_1 + \cdots + d_{N-1} - n - 1)$. Hence, intersection theory on projective space gives a formula for the genus of $X$ entirely in terms of the various $d_i$. Specifically, $$2g - 2 = d_1\cdots d_{N-1} (d_1 + \cdots + d_{N-1} - N - 1)$$

$\textbf{Question:}$ Suppose $X$ is a nonsingular curve in $\mathbb{P}^N$, and $X = V(F_1, \cdots, F_m)$ is not a complete intersection. Can one get a similarly simple formula for the genus of $X$, perhaps entirely in terms of the degrees of the $F_i$? Is this too much to ask? I can't even simply describe the canonical divisor.

`$d\leq \text{max}(\text{deg}(F_i))^{N-1}$`

. For a generic linear projection of $X$ to`$\mathbb{P}^2$`

, the image is an at worst nodal curve whose geometric genus is less than the arithmetic genus, i.e., $g(X) \leq (d-1)(d-2)/2$. Certainly if you impose further restrictions on $X$ or the equations $F_i$, then you can improve this. However, this is sharp for plane curves. $\endgroup$ – Jason Starr Nov 26 '12 at 14:441more comment