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Algebra Seminar

POT 745
Speaker(s) / Presenter(s):
David Stapleton, University of Michigan
Title. Smooth limits of plane curves and Markov numbers
Abstract. When can we guarantee that smooth proper limits of plane curves are still plane curves? Said a different way --- When is the locus of degree d plane curves closed in the (noncompact) moduli space of smooth genus g curves? It is relatively easy to see that if d>1, then d must be prime. Mori suggests that this may be enough in higher dimensions. Interestingly, in low dimensions, this is not sufficient. For example, Griffin constructed explicit families of quintic plane curves with a smooth limit that is not a quintic plane curve. In this talk we propose the following conjecture: Smooth proper limits of plane curves of degree d are always plane curves if d is prime and d is not a Markov number. We discuss the motivation and evidence for this conjecture which come from Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane.
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