Insalata di Matematica "Coincident root loci and the moduli space of rational elliptic surfaces"

Elliptic fibrations are strongly characterized by their singular fibers. We analyze the moduli spaces of elliptic surfaces over the projective line \mathbb{P}^1 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 \mathbb{P}^d be the space parametrizing homogeneous degree d polynomials in two variables: what happens if the roots of the polynomials collapse? Given a partition \sigma of d, how is the locus in \mathbb{P}^d corresponding to polynomials having roots with the multiplicities prescribed by \sigma?

 

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