Abstract
Given an arbitrary compact Riemann surface X, of genus g, wouldn’t it be handy if we had a holomorphic function E: X × X → ¢ such that E(x,y) = 0 if and only if x = y? Although such a function doesn’t exist, it turns out that it “almost” does! To understand part of the problem and how to fix it, let’s look at the simplest case: Example. Let X = IP1. The function x-y works on IP1-{∞} but not on all of IP1. So consider instead the “differential”:
where \( \sqrt {dx} ,\sqrt {dy} \) are defined as follows.
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© 2007 Birkhäuser Boston
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Mumford, D. (2007). The Prime Form E(x,y).. In: Tata Lectures on Theta II. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4578-6_13
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DOI: https://doi.org/10.1007/978-0-8176-4578-6_13
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-4569-4
Online ISBN: 978-0-8176-4578-6
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