4
$\begingroup$

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

  • 0
    @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.2012-08-02

2 Answers 2

1

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

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.