Abstract: Elliptic fibrations are strongly characterised by their singular fibers. We analyse the moduli spaces of elliptic surfaces over the projective line P1 and construct a stratification of these spaces in terms of the singular fibers that the surfaces have. In order to do that, we need to investigate a very natural problem: let Pd be the space parametrising homogeneous degree d polynomials in two variables; what happens if the roots of the polynomials collapse? Given a partition λ of d, how is the locus in Pd corresponding to polynomials having roots with the multiplicities prescribed by λ?
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