In the complex projective space, one considers the algebraic surface $z_0^m+z_1^m+z_2^m+z_3^m=0$ and restricts to $S$ the projection to the plane $z_3=0$. So one obtains $S$, an $m-$fold branched covering of the complex projective plane with the plane algebraic curve $C$ of equation $z_0^m+z_1^m+z_2^m=0$ as its branch locus of order $m-1$. Let $\chi (S)$ be the Euler number of this manifold.

I know that $\chi(S)=m \chi(\mathbb{P}^2)-(m-1)\chi(C)$.

Can you give me the general formula of which the previous is a special case and in which book I can find it, please!


1 Answer 1


Let $X$ be a projective variety and $Y$ a divisor. Then $\chi(X) = \chi(Y) + \chi(X\setminus Y)$ where $\chi$ denotes the topological Euler characteristic; see Proposition $7.16$ of $3264$ & All That by Eisenbud and Harris.

Suppose $Z$ is an $m$-sheeted cyclic cover of $X$ branched along $Y$. That is, there is a map $\pi : Z \to X$ and a divisor $Y'$ of $Z$, such that $\pi$ maps $Y'$ isomorphically onto $Y$, and $\pi_{Z\setminus Y'} : Z\setminus Y' \to X\setminus Y$ is an $m$-sheeted covering map. Then, as the Euler characteristic is multiplicative under covers, we have

\begin{align*} \chi(Z) &= \chi(Y') + \chi(Z\setminus Y')\\ &= \chi(Y) + m\chi(X\setminus Y)\\ &= m\chi(Y) + m\chi(X\setminus Y) - (m-1)\chi(Y)\\ &= m\chi(X) - (m-1)\chi(Y). \end{align*}

In this case, $S$ is a cyclic cover associated to the line bundle $\mathcal{O}(m) = \mathcal{O}(1)^{\otimes m}$ on $\mathbb{P}^2$.


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