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 $m_i$

  • $\begingroup$ 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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy