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In most papers on elliptic surfaces (at least among those I am aware of) the condition "there is at least one singular fiber" is to exclude elliptic surfaces that are a product or to enforce similar p …

For the case with section and $q=0$ see http://www.math.colostate.edu/~miranda/preprints/weierstrassfibrations.pdf A similar construction should work in the case (with section, $q$ fixed and $p_g$ su …

I do not believe you have equality in general. I sketch a counterexample below, which is a geometric version of the fact that if $E/K$ is an elliptic curves such that the quadratic twist $E^{(d)}/K$ h …

I will sketch a counterexample for the modified question. The idea behind the construction is similar to the counterexample for the original question. Only the geometric details of this construction a …

The modular elliptic surfaces are quite rare. E.g., the Mordell-Weil group is finite and the Picard number of the surface equals $h^{1,1}$ (see Shioda's paper). Such elliptic surfaces are called extre …

Consider first an elliptic fibration with a section over $\mathbb{P}^1$. (In this case none of the singular fibers are multiples of smooth curves.) Assume that the minimal discriminant has degree $12 …