# Genus of curves embedded into some projective space

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 $f_1,\ldots,f_{n-1}$?

• Would you consider the Hilbert polynomial as "easy data"? – curious Aug 2 '12 at 7:47
• There is Exercise II 8.4 (g) in Hartshorne which says that, for $n=3$, if $C$ is nonsingular $f_1$ and $f_2$ define nonsingular hypersurfaces of degrees $d_1$ and $d_2$, you have $p_g=\tfrac 12 d_1 d_2(d_1+d_2-4) + 1$. This might all be generalized, but what are you willing to assume? You are clearly willing to assume that $C$ is a complete intersection, but what about singularities? – Jesko Hüttenhain Aug 2 '12 at 7:54
• @curious. I think I consider the Hilbert polynomial as "easy data". To rattle. I want to take care of singularities too. Let's keep them ordinary double if that helps. This way $C$ is a stable curve. The usual Plucker formula gives a formula in this case. – Harry Aug 2 '12 at 7:56

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.