Genus of curves embedded into some projective space

Question

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$.

Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let $f_1,\ldots,f_{n-1}$ be homogenous polynomials in $k[x_0,x_1,\ldots,x_n]$ and let $C$ be the curve defined by $f_1=0,\ldots,f_{n-1}=0$ in $\mathbf{P}^n$. Is there a formula to compute the genus of $C$ in terms of easy data depending only on $f_1,\ldots,f_{n-1}$?

Answer 1

This paper does it for you (also available here).

Answer 2

If you consider the Hilbert polynomial $P_C$ "easy data", then you can use the formula $$p_a(C) = (-1)^{\dim(C)} (P_C(0)-1)$$ to calculate the arithmetic genus. See, e.g. Hartshorne Exercise I.7.2.