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DTSTAMP:20170417T115523Z
DTSTART;TZID=America/Los_Angeles:20170417T160000
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SUMMARY:Student Algebraic Geometry Seminar: Geometry of the moduli space of curves of genus \\(g\\)
UID:108750-ucb-events-calendar@berkeley.edu
ORGANIZER;CN="UC Berkeley Calendar Network":
LOCATION:891 Evans Hall
DESCRIPTION:Ritvik Ramkumar\, UC Berkeley\n\nGiven a variety \\(X\\) it's a basic question to ask if it's rational i.e. admits a birational map \\( P^n \\to X \\) for some \\(n\\). For curves and surfaces there are explicit criterion that determine when a variety is rational. Answering this question in higher dimensions is much more difficult. In light of this\, it's easier to ask for a weaker question: Does there exist a dominant rational map \\(P^n\\to X\\). In this case\, \\(X\\) is said to be unirational. Over the complex numbers\, every unirational curve or surface is rational. For curves this is a consequence of the Riemann-Hurwitz Formula.\n\nA very important geometric object is \\(M_g\\)\, the moduli space of curves of genus \\(g\\). We can ask for what \\(g\\)\, if any\, is \\(M_g\\) rational or unirational. In 1915\, Severi proved that \\(M_g\\) is unirational for \\(g\\leq 10\\) and conjectured that it's unirational for all \\(g\\). However\, in 1987\, it was shown that \\(M_g\\) is of general type if \\(g\\geq 24\\). In this talk I will review classical examples of unirational varieties\, outline a modern proof of Severi's result and describe related conjectures and results. I will finish by describing how the geometry of \\(M_g\\) changes when \\(g\\geq 11\\).
URL:http://events.berkeley.edu/index.php/calendar/sn/pubaff.html?event_ID=108750&view=preview
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