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I'm looking for a small research problem an undergraduate would be capable of after taking just an abstract algebra course, introductory algebraic geometry (at level of Miles Reid's book and Ideals, Varieties & Algorithms), and a course in number theory. Is there a website that would have a decent listing, or possibly a book one can recommend that may have small open research problems?

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    $\begingroup$ Open problems in algebraic geometry accessible to an advanced undergraduate seem scarce enough that I would keep any I thought of as trade secrets. And at this moment I can't think of anything else mathematical I would keep as a trade secret! $\endgroup$
    – Alexander Woo
    Commented Jun 15, 2010 at 17:03

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Your question is missing a crucial word in the first sentence (capable of ...). Is the missing word "understanding" or "solving"?

Anyway, here is a problem: Find the maximum number of points of a curve of genus $g$ over $\mathbb{F}_q$, for some values of $g,q$ for which this number is not known (check for values at http://www.manypoints.org/ )

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  • $\begingroup$ @Felipe, I will be amazed if someone who knows algebraic geometry from the given sources can deal with it over a field not algebraically closed. Also, Reid's book avoids Riemann-Roch, so genus will be tough to use algebraically, let alone over finite fields. There's always plane curves, but undergraduate research in these areas seems a dubious idea. Perhaps another crucial word is missing: elementary number theory or algebraic number theory? $\endgroup$
    – Boyarsky
    Commented Jun 15, 2010 at 18:11

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