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Is there any proof for the following statement? It has been used as trivial fact in the one of papers of Edward Witten

Let $\Sigma$ be a compact connected Riemann surface as orbifold, with marked points $x_1,\cdots x_p$ of multiplicities $m_1,\cdots,m_p$ then is there always exists an elliptic surface $X$ with $\chi(X)>0$ over $\Sigma$, ($\pi:X\to \Sigma$) such that the general fibers are smooth elliptic curve and over the marked points $x_i$ one has multiple fibers $\pi^{-1}(x_i)$ of multiplicity $m_i$

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  • $\begingroup$ Is there any condition on possible singular fibers away from the marked points? $\endgroup$
    – Will Sawin
    Jun 7, 2017 at 2:46
  • $\begingroup$ No, there is no assumption on singular fibers $\endgroup$
    – user21574
    Jun 7, 2017 at 2:49

1 Answer 1

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Let us start from any elliptic surface $X_0 \longrightarrow \Sigma$ with $\chi(X_0) >0$ and whose fibre at the point $x_i$ is of type $I_0$ (i.e., smooth) for all $i \in \{1, \ldots, p\}$. For instance, a general elliptic fibration over $\Sigma$ without multiple fibres satisfies these requirements.

Then with a sequence of $p$ logarithmic transformations of order $m_i$ and centered at $x_i$ we can construct a new elliptic surface $X \longrightarrow \Sigma$ such that its fibre at the point $x_i$ is of type ${}_{m_i}I_0$, i.e. a smooth elliptic curve with multiplicity $m_i$.

Logarithmic transformations do not change the Euler number, so $$\chi(X) = \chi(X_0)>0$$ and we are done.

See Barth, Peters, Van de Ven's book Compact Complex Surfaces, V.13 for more details.

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