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Let $\mathcal{L}=L_1\cup \ldots \cup L_k$ be an arrangement of lines in $\mathbb{P}^2$ with $k>2$ and $t_k=0$ ($t_r$ is defined to be the number of $r$-fold points of the arrangement for $r\in \mathbb{N}$). For any natural number $n>1$ we may associate with $\mathcal{L}$ the following function field $$F_{\mathcal{L},n}:=\mathbb{C}(z_1/z_0,z_2/z_0)\left((l_2/l_1)^{1/n},\ldots, (l_k/l_1)^{1/n}\right), $$ where $z_0, z_1, z_2$ are the homogeneous coordinates on $\mathbb{P}^2$ and each $l_i$ is a linear form associated with $L_i$.

In Hirzebruch's paper "Arrangements of lines and algebraic surfaces" he states that "The function field ($F_{\mathcal{L},n}$) determines an algebraic surface $X$ with normal singularities which ramifies over the plane with the arrangement ($\mathcal{L}$) as the locus of ramification."

I have two questions regarding this construction:

  1. I understand that the given function field would be associated with a birational equivalence class of an algebraic surface, but how do we know immediately that this function field determines an algebraic surface with normal singularities, as it seems we may just as well associate with it any smooth surface in the same birational equivalence class. So I guess what I'm really asking is given an explicit description of a function field (or more specifically an abelian extension of the function field of P^2) how do we go about associating with it an explicit (possibly singular) algebraic surface?

  2. May we derive from this description of the function field an explicit embedding of the surface in some ambient variety (say $\mathbb{P}^N$)?

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up vote 2 down vote accepted

Regarding 1: If $X$ is a variety with function field $K$, $L$ a finite separable extension of $K$, we can construct a normal variety $Y$, "normalization of $X$ in $L$", with function field $L$ and a finite map $Y\to X$. This is done as follows: for any affine open $X_i = Spec(R)$ in $X$, take $Y_i = Spec(S)$ where $S$ is the integral closure of $R$ in $L$, and then glue these $Y_i$ to obtain $Y$. In your situation, $X = \mathbb{P}^2$, $K = \mathbb{C}(z_1/z_0, z_2/z_0)$, $L$ is the field you described and $Y$ is the surface you need.

Regarding 2: I cannot say anything explicit, but if $f:Y\to X$ is a finite map and $L$ is an ample line bundle on $X$, then $f^* L$ is an ample line bundle on $Y$. This gives you a projective embedding of $Y$ (up to Veronese). Sorry I can't be more helpful here...

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